{
"query": {
"display": "$$y^{^{\\prime\\prime}}-y^{^{\\prime}}+121y=11\\sin\\left(11t\\right)$$",
"symbolab_question": "ODE#y^{\\prime \\prime }-y^{\\prime }+121y=11\\sin(11t)"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "ODE",
"subTopic": "ConstCoeffLinearNonHomogeneous",
"default": "y=e^{\\frac{t}{2}}(c_{1}\\cos(\\frac{\\sqrt{483}t}{2})+c_{2}\\sin(\\frac{\\sqrt{483}t}{2}))+\\cos(11t)",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=11\\sin\\left(11t\\right):{\\quad}y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)+\\cos\\left(11t\\right)$$",
"input": "y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=11\\sin\\left(11t\\right)",
"steps": [
{
"type": "interim",
"title": "Resolver EDO lineales:$${\\quad}y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)+\\cos\\left(11t\\right)$$",
"input": "y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=11\\sin\\left(11t\\right)",
"steps": [
{
"type": "definition",
"title": "Ecuación diferencial no homogénea de segundo orden lineal con coeficientes constantes",
"text": "Una EDO lineal, no homogénea de segundo orden tiene la siguiente forma $$ay''+by'+cy=g\\left(x\\right)$$"
},
{
"type": "step",
"primary": "La solución general para $$a\\left(x\\right)y''+b\\left(x\\right)y'+c\\left(x\\right)y=g\\left(x\\right)$$ se puede escribir como<br/>$$y=y_h+y_p$$<br/>$$y_h$$ es la solución para la EDO homogenea $$a\\left(x\\right)y''+b\\left(x\\right)y'+c\\left(x\\right)y=0$$<br/>$$y_p$$, la solución particular, es cualquier función que satisface la ecuación no homogenea "
},
{
"type": "interim",
"title": "Hallar $$y_h$$ resolviendo $$y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=0:{\\quad}y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)$$",
"input": "y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=0",
"steps": [
{
"type": "definition",
"title": "Ecuación diferencial homogénea lineal de segundo orden con coeficientes constantes",
"text": "Una EDO homogenea, lineal de segundo orden tiene la siguiente forma $$ay''+by'+cy=0$$"
},
{
"type": "step",
"primary": "Para una ecuación $$ay''+by'+cy=0$$, asumir una solución con la forma $$e^{γt}$$",
"secondary": [
"Re escribir la ecuación con $$y=e^{γt}$$"
],
"result": "\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime\\prime}}-\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime}}+121e^{γt}=0"
},
{
"type": "interim",
"title": "Simplificar $$\\left(\\left(e^{γt}\\right)\\right)^{\\prime\\prime}-\\left(\\left(e^{γt}\\right)\\right)^{\\prime}+121e^{γt}=0:{\\quad}e^{γt}\\left(γ^{2}-γ+121\\right)=0$$",
"steps": [
{
"type": "step",
"result": "\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime\\prime}}-\\left(\\left(e^{γt}\\right)\\right)^{^{\\prime}}+121e^{γt}=0"
},
{
"type": "interim",
"title": "$$\\left(e^{γt}\\right)^{\\prime\\prime}=γ^{2}e^{γt}$$",
"input": "\\left(e^{γt}\\right)^{\\prime\\prime}",
"steps": [
{
"type": "interim",
"title": "$$\\left(e^{γt}\\right)^{\\prime}=e^{γt}γ$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}e^{γt}\\left(γt\\right)^{\\prime}$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"result": "=e^{γt}\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=γt$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γt\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(γt\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=γt$$",
"result": "=e^{γt}\\left(γt\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcDDfSBXef41nZ6zbP7ViaAosjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5PvoGisVaN+BwjjtpPeRZCLDrbw8lc2jRiiaaodUFzB+z/t6iERavDspCBDo0aAo/Mx5f3j1BGXVguN+LdrRlNX8G+nPzP8wZMNixoBFUjjdQ=="
}
},
{
"type": "interim",
"title": "$$\\left(γt\\right)^{\\prime}=γ$$",
"input": "\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γt^{^{\\prime}}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$t^{\\prime}=1$$",
"result": "=γ\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=γ",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7stv4XdlIkdSrTZc5soZLGCENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tSbX/i/cqXdrp84USJNBCUvvRCDs4D3rcIVpx7C72k9c"
}
},
{
"type": "step",
"result": "=e^{γt}γ"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=\\left(e^{γt}γ\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(e^{γt}γ\\right)^{\\prime}=γ^{2}e^{γt}$$",
"input": "\\left(e^{γt}γ\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γ\\left(e^{γt}\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}e^{γt}\\left(γt\\right)^{\\prime}$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"result": "=e^{γt}\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=γt$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γt\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(γt\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=γt$$",
"result": "=e^{γt}\\left(γt\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcDDfSBXef41nZ6zbP7ViaAosjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5PvoGisVaN+BwjjtpPeRZCLDrbw8lc2jRiiaaodUFzB+z/t6iERavDspCBDo0aAo/Mx5f3j1BGXVguN+LdrRlNX8G+nPzP8wZMNixoBFUjjdQ=="
}
},
{
"type": "interim",
"title": "$$\\left(γt\\right)^{\\prime}=γ$$",
"input": "\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γt^{^{\\prime}}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$t^{\\prime}=1$$",
"result": "=γ\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=γ",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7stv4XdlIkdSrTZc5soZLGCENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tSbX/i/cqXdrp84USJNBCUvvRCDs4D3rcIVpx7C72k9c"
}
},
{
"type": "step",
"result": "=γe^{γt}γ"
},
{
"type": "interim",
"title": "Simplificar $$γe^{γt}γ:{\\quad}γ^{2}e^{γt}$$",
"input": "γe^{γt}γ",
"result": "=γ^{2}e^{γt}",
"steps": [
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$γγ=\\:γ^{1+1}$$"
],
"result": "=e^{γt}γ^{1+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Sumar: $$1+1=2$$",
"result": "=e^{γt}γ^{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7riofKsRQ/FkpZv0BW2pxMd6GQqufR6tr2vPxOUv7H+/BWItNlNCsjK5QfFqGTa8umx4rCXhbsN+br+uOYP22UU3kCh3oevUunZ7/b0qFKBQzgpyv32CMKtRLUB0OSZdsvVQ3sHiLuCyKO/MYghjk7Q=="
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=γ^{2}e^{γt}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "γ^{2}e^{γt}-\\left(e^{γt}\\right)^{^{\\prime}}+121e^{γt}=0"
},
{
"type": "interim",
"title": "$$\\left(e^{γt}\\right)^{\\prime}=e^{γt}γ$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"steps": [
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}e^{γt}\\left(γt\\right)^{\\prime}$$",
"input": "\\left(e^{γt}\\right)^{\\prime}",
"result": "=e^{γt}\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=e^{u},\\:\\:u=γt$$"
],
"result": "=\\left(e^{u}\\right)^{^{\\prime}}\\left(γt\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"input": "\\left(e^{u}\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\left(e^{u}\\right)^{\\prime}=e^{u}$$",
"result": "=e^{u}"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cVPV4zUFISiTd+fVX6xXsrmsNRuddYPgZ8cGsLVhNNRQsU0KegSjwRVV1JfeZUqosl5PTRzFd2J0fcq0+01bpNW4Yoa9OGLIL+u1HBPyzhvQzhwSHylow7u2/8ADWpoHsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "step",
"result": "=e^{u}\\left(γt\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=γt$$",
"result": "=e^{γt}\\left(γt\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gvl9rbrp9vJgB3DE3RVCcDDfSBXef41nZ6zbP7ViaAosjvX7KVUO/AeCFSId4S33iWw9g5uXzmS5KX5zIzOHZXiX35dQ/h01lIvxamZtt5PvoGisVaN+BwjjtpPeRZCLDrbw8lc2jRiiaaodUFzB+z/t6iERavDspCBDo0aAo/Mx5f3j1BGXVguN+LdrRlNX8G+nPzP8wZMNixoBFUjjdQ=="
}
},
{
"type": "interim",
"title": "$$\\left(γt\\right)^{\\prime}=γ$$",
"input": "\\left(γt\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=γt^{^{\\prime}}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$t^{\\prime}=1$$",
"result": "=γ\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=γ",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7stv4XdlIkdSrTZc5soZLGCENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoUpO3zWZspTvnswNQKdz3tSbX/i/cqXdrp84USJNBCUvvRCDs4D3rcIVpx7C72k9c"
}
},
{
"type": "step",
"result": "=e^{γt}γ"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "γ^{2}e^{γt}-e^{γt}γ+121e^{γt}=0"
},
{
"type": "step",
"primary": "Factorizar $$e^{γt}$$",
"result": "e^{γt}\\left(γ^{2}-γ+121\\right)=0"
}
],
"meta": {
"interimType": "Generic Simplify Specific 1Eq"
}
},
{
"type": "step",
"result": "e^{γt}\\left(γ^{2}-γ+121\\right)=0"
},
{
"type": "interim",
"title": "Resolver $$e^{γt}\\left(γ^{2}-γ+121\\right)=0:{\\quad}γ=\\frac{1}{2}+i\\frac{\\sqrt{483}}{2},\\:γ=\\frac{1}{2}-i\\frac{\\sqrt{483}}{2}$$",
"input": "e^{γt}\\left(γ^{2}-γ+121\\right)=0",
"steps": [
{
"type": "step",
"primary": "Ya que $$e^{γt}\\ne\\:0$$, resolver $$e^{γt}\\left(γ^{2}-γ+121\\right)=0$$<br/> es equivalente a resolver la ecuación cuadrática $$γ^{2}-γ+121=0$$",
"result": "γ^{2}-γ+121=0"
},
{
"type": "interim",
"title": "Resolver con la fórmula general para ecuaciones de segundo grado:",
"input": "γ^{2}-γ+121=0",
"result": "{γ}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:1\\cdot\\:121}}{2\\cdot\\:1}",
"steps": [
{
"type": "definition",
"title": "Formula general para ecuaciones de segundo grado:",
"text": "Para una ecuación de segundo grado de la forma $$ax^2+bx+c=0$$ las soluciones son <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$"
},
{
"type": "step",
"primary": "Para $${\\quad}a=1,\\:b=-1,\\:c=121$$",
"result": "{γ}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:1\\cdot\\:121}}{2\\cdot\\:1}"
}
],
"meta": {
"interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq",
"gptData": "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"
}
},
{
"type": "interim",
"title": "Simplificar $$\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:1\\cdot\\:121}:{\\quad}\\sqrt{483}i$$",
"input": "\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:1\\cdot\\:121}",
"result": "{γ}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{483}i}{2\\cdot\\:1}",
"steps": [
{
"type": "interim",
"title": "$$\\left(-1\\right)^{2}=1$$",
"input": "\\left(-1\\right)^{2}",
"steps": [
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\left(-a\\right)^{n}=a^{n},\\:$$si $$n$$ es par",
"secondary": [
"$$\\left(-1\\right)^{2}=1^{2}$$"
],
"result": "=1^{2}"
},
{
"type": "step",
"primary": "Aplicar la regla $$1^{a}=1$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g="
}
},
{
"type": "interim",
"title": "$$4\\cdot\\:1\\cdot\\:121=484$$",
"input": "4\\cdot\\:1\\cdot\\:121",
"steps": [
{
"type": "step",
"primary": "Multiplicar los numeros: $$4\\cdot\\:1\\cdot\\:121=484$$",
"result": "=484"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xF9rlQkuAOrRzKTf6eQs7c0qtp6QqZdTxND6Y0QEWHVwkKGJWEPFPk38sdJMsyPI5v4BiYr/j5RJR51+n8IwSMRQ2Rwca5Tm02OJRCslEJ9XABBR49BhKfDbLf0QHU3n"
}
},
{
"type": "step",
"result": "=\\sqrt{1-484}"
},
{
"type": "step",
"primary": "Restar: $$1-484=-483$$",
"result": "=\\sqrt{-483}"
},
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\sqrt{-a}=\\sqrt{-1}\\sqrt{a}$$",
"secondary": [
"$$\\sqrt{-483}=\\sqrt{-1}\\sqrt{483}$$"
],
"result": "=\\sqrt{-1}\\sqrt{483}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Aplicar las propiedades de los numeros imaginarios: $$\\sqrt{-1}=i$$",
"result": "=\\sqrt{483}i",
"meta": {
"practiceLink": "/practice/complex-numbers-practice",
"practiceTopic": "Expand FOIL"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"primary": "Separar las soluciones",
"result": "{γ}_{1}=\\frac{-\\left(-1\\right)+\\sqrt{483}i}{2\\cdot\\:1},\\:{γ}_{2}=\\frac{-\\left(-1\\right)-\\sqrt{483}i}{2\\cdot\\:1}"
},
{
"type": "interim",
"title": "$$γ=\\frac{-\\left(-1\\right)+\\sqrt{483}i}{2\\cdot\\:1}:{\\quad}\\frac{1}{2}+i\\frac{\\sqrt{483}}{2}$$",
"input": "\\frac{-\\left(-1\\right)+\\sqrt{483}i}{2\\cdot\\:1}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla $$-\\left(-a\\right)=a$$",
"result": "=\\frac{1+\\sqrt{483}i}{2\\cdot\\:1}"
},
{
"type": "step",
"primary": "Multiplicar los numeros: $$2\\cdot\\:1=2$$",
"result": "=\\frac{1+\\sqrt{483}i}{2}"
},
{
"type": "interim",
"title": "Reescribir $$\\frac{1+\\sqrt{483}i}{2}$$ en la forma binómica: $$\\frac{1}{2}+\\frac{\\sqrt{483}}{2}i$$",
"input": "\\frac{1+\\sqrt{483}i}{2}",
"steps": [
{
"type": "step",
"primary": "Aplicar las propiedades de las fracciones: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$",
"secondary": [
"$$\\frac{1+\\sqrt{483}i}{2}=\\frac{1}{2}+\\frac{\\sqrt{483}i}{2}$$"
],
"result": "=\\frac{1}{2}+\\frac{\\sqrt{483}i}{2}"
}
],
"meta": {
"interimType": "Rewrite In Complex Form Title 2Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYtJEyuaCK6wsxSwODyDWemAnVQZ9jTQhWaoOGaFYelMKfAu5u/TBlzVG5qXgF9PAhyjetd55DYlveZzsS8XHZnp6pfF1z6umzUJTJvt+ojYZY6GiCEoDrrgp/ch/8HXnpRtNnbfnlsK110uSEScdJPc+ec36N/WH1h1pArd7emylpAqBUrpyBwYHc4PzvTKuNh4kfSKf5t3Dn7AAWQ1m9xY="
}
},
{
"type": "step",
"result": "=\\frac{1}{2}+\\frac{\\sqrt{483}}{2}i"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/7pkeZ9PvpPR5geH3JsPMytyDxpoo2WjP0dOy06Ngp9oHpYayvDNk2OAeHm77brGCUCWbkwGOY7PqKo3U/JLJSQRqTjrUDd23baZ6wFVd3VOypTIYDWwGawep0wQDTvEc93lThIt3Q/3wGfwYj1G2herdkx4fh/64hGjtYuWxjb1FSLpfUk3vbW/M6oOdQCaWNkdcQsCCvOiAG92qWEuDbCI2sSeA74029n2yo277ZU="
}
},
{
"type": "interim",
"title": "$$γ=\\frac{-\\left(-1\\right)-\\sqrt{483}i}{2\\cdot\\:1}:{\\quad}\\frac{1}{2}-i\\frac{\\sqrt{483}}{2}$$",
"input": "\\frac{-\\left(-1\\right)-\\sqrt{483}i}{2\\cdot\\:1}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla $$-\\left(-a\\right)=a$$",
"result": "=\\frac{1-\\sqrt{483}i}{2\\cdot\\:1}"
},
{
"type": "step",
"primary": "Multiplicar los numeros: $$2\\cdot\\:1=2$$",
"result": "=\\frac{1-\\sqrt{483}i}{2}"
},
{
"type": "interim",
"title": "Reescribir $$\\frac{1-\\sqrt{483}i}{2}$$ en la forma binómica: $$\\frac{1}{2}-\\frac{\\sqrt{483}}{2}i$$",
"input": "\\frac{1-\\sqrt{483}i}{2}",
"steps": [
{
"type": "step",
"primary": "Aplicar las propiedades de las fracciones: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$",
"secondary": [
"$$\\frac{1-\\sqrt{483}i}{2}=\\frac{1}{2}-\\frac{\\sqrt{483}i}{2}$$"
],
"result": "=\\frac{1}{2}-\\frac{\\sqrt{483}i}{2}"
}
],
"meta": {
"interimType": "Rewrite In Complex Form Title 2Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYps4fmygri5eHQCBqKksYronVQZ9jTQhWaoOGaFYelMKfAu5u/TBlzVG5qXgF9PAhyjetd55DYlveZzsS8XHZnp6pfF1z6umzUJTJvt+ojYZY6GiCEoDrrgp/ch/8HXnpeGBeHRaI+atKxgWyR1gcL8+ec36N/WH1h1pArd7emylpAqBUrpyBwYHc4PzvTKuNh4kfSKf5t3Dn7AAWQ1m9xY="
}
},
{
"type": "step",
"result": "=\\frac{1}{2}-\\frac{\\sqrt{483}}{2}i"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XXvJ86m2ThoGD7P681gRfCtyDxpoo2WjP0dOy06Ngp9oHpYayvDNk2OAeHm77brGCUCWbkwGOY7PqKo3U/JLJSQRqTjrUDd23baZ6wFVd3X45Pu3hs6dyTvLPVvT6J0Xc93lThIt3Q/3wGfwYj1G2herdkx4fh/64hGjtYuWxjbNrMcvNkQJ7akK/12zrGEiWNkdcQsCCvOiAG92qWEuDbCI2sSeA74029n2yo277ZU="
}
},
{
"type": "step",
"primary": "Las soluciones a la ecuación de segundo grado son: ",
"result": "γ=\\frac{1}{2}+i\\frac{\\sqrt{483}}{2},\\:γ=\\frac{1}{2}-i\\frac{\\sqrt{483}}{2}"
}
],
"meta": {
"interimType": "Generic Solve Title 1Eq"
}
},
{
"type": "step",
"result": "γ=\\frac{1}{2}+i\\frac{\\sqrt{483}}{2},\\:γ=\\frac{1}{2}-i\\frac{\\sqrt{483}}{2}"
},
{
"type": "step",
"primary": "Para dos raices complejas $$γ_{1}\\ne\\:γ_{2}$$, donde $$γ_{1}=\\alpha+i\\:\\beta,\\:γ_{2}=\\alpha-i\\:\\beta\\:$$<br/>la solución general toma la forma:$${\\quad}y=e^{\\alpha\\:t}\\left(c_{1}\\cos\\left(\\beta\\:t\\right)+c_{2}\\sin\\left(\\beta\\:t\\right)\\right)$$",
"result": "e^{\\frac{1}{2}t}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}}{2}t\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}}{2}t\\right)\\right)"
},
{
"type": "step",
"primary": "Simplificar",
"result": "y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)"
}
],
"meta": {
"solvingClass": "ODE",
"interimType": "Generic Find By Solving Title 2Eq"
}
},
{
"type": "interim",
"title": "Encontrar $$y_{p}$$ que satisfaga $$y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=11\\sin\\left(11t\\right):{\\quad}y=\\cos\\left(11t\\right)$$",
"steps": [
{
"type": "step",
"primary": "Para la parte no homogénea $$g\\left(x\\right)=11\\sin\\left(11t\\right)$$, asumir una solución con la forma: $$y=a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)$$"
},
{
"type": "step",
"result": "\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{^{\\prime\\prime}}-\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{^{\\prime}}+121\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)=11\\sin\\left(11t\\right)"
},
{
"type": "interim",
"title": "Simplificar $$\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime\\prime}-\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime}+121\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)=11\\sin\\left(11t\\right):{\\quad}-11a_{0}\\cos\\left(11t\\right)+11a_{1}\\sin\\left(11t\\right)=11\\sin\\left(11t\\right)$$",
"steps": [
{
"type": "interim",
"title": "$$\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime\\prime}=-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)$$",
"input": "\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime\\prime}",
"steps": [
{
"type": "interim",
"title": "$$\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}=a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)$$",
"input": "\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la suma/diferencia: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\left(a_{0}\\sin\\left(11t\\right)\\right)^{^{\\prime}}+\\left(a_{1}\\cos\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(a_{0}\\sin\\left(11t\\right)\\right)^{\\prime}=a_{0}\\cos\\left(11t\\right)\\cdot\\:11$$",
"input": "\\left(a_{0}\\sin\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=a_{0}\\left(\\sin\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\sin\\left(11t\\right)\\right)^{\\prime}",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\sin\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\sin\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"input": "\\left(\\sin\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"result": "=\\cos\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przbRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19InMhwrDgcALtDbcjUqULjO0e5byrQDQVCXUD0vH/fvOdy0Sv9DakosoaSdvz3y7/jiRKGGP17LApxwbbz9SLndow=="
}
},
{
"type": "step",
"result": "=\\cos\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=11t$$",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przdXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcSTIe30CBuWmP1+uyGYZgZDHt0FLgpzbmBtaZEH6JjLtbA+zX4bD3u3gx65o2NJhMpxXNauH4WBlwhMiNwcYlXboN1Jhd8xIGBjH0OZzd+PbCI2sSeA74029n2yo277ZU="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=a_{0}\\cos\\left(11t\\right)\\cdot\\:11"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "interim",
"title": "$$\\left(a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}=-11a_{1}\\sin\\left(11t\\right)$$",
"input": "\\left(a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=a_{1}\\left(\\cos\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\cos\\left(11t\\right)\\right)^{\\prime}",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\cos\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\cos\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"input": "\\left(\\cos\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"result": "=-\\sin\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyLRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19In4/yNG6RtaMeFUKXMVc75z2DpznShoZau34eGycg+7G3mVcoSp6dsHpd9bDcXuJ0pz2gvuHI/drUkZ7NmziNx4SS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "step",
"result": "=-\\sin\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=11t$$",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyNXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcOXl4x9OXOWFHQY9rAT64IGgK/z24D94dcZ1AqLbMwBPC30sSftAIFS6Qkpy19IkqbEmKQhIYyRwegIjh1uziVbUYzbseyJBZAzad0tEQA4ompXFf3SOUx+H18qfp3MLg="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=a_{1}\\left(-\\sin\\left(11t\\right)\\cdot\\:11\\right)"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=-11a_{1}\\sin\\left(11t\\right)",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)\\right)^{\\prime}=-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)$$",
"input": "\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la suma/diferencia: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11\\right)^{^{\\prime}}-\\left(11a_{1}\\sin\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11\\right)^{\\prime}=-121a_{0}\\sin\\left(11t\\right)$$",
"input": "\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=a_{0}\\cdot\\:11\\left(\\cos\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\cos\\left(11t\\right)\\right)^{\\prime}",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\cos\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\cos\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"input": "\\left(\\cos\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"result": "=-\\sin\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyLRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19In4/yNG6RtaMeFUKXMVc75z2DpznShoZau34eGycg+7G3mVcoSp6dsHpd9bDcXuJ0pz2gvuHI/drUkZ7NmziNx4SS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "step",
"result": "=-\\sin\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=11t$$",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyNXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcOXl4x9OXOWFHQY9rAT64IGgK/z24D94dcZ1AqLbMwBPC30sSftAIFS6Qkpy19IkqbEmKQhIYyRwegIjh1uziVbUYzbseyJBZAzad0tEQA4ompXFf3SOUx+H18qfp3MLg="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=a_{0}\\cdot\\:11\\left(-\\sin\\left(11t\\right)\\cdot\\:11\\right)"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=-121a_{0}\\sin\\left(11t\\right)",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "interim",
"title": "$$\\left(11a_{1}\\sin\\left(11t\\right)\\right)^{\\prime}=121a_{1}\\cos\\left(11t\\right)$$",
"input": "\\left(11a_{1}\\sin\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11a_{1}\\left(\\sin\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\sin\\left(11t\\right)\\right)^{\\prime}",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\sin\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\sin\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"input": "\\left(\\sin\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"result": "=\\cos\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przbRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19InMhwrDgcALtDbcjUqULjO0e5byrQDQVCXUD0vH/fvOdy0Sv9DakosoaSdvz3y7/jiRKGGP17LApxwbbz9SLndow=="
}
},
{
"type": "step",
"result": "=\\cos\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=11t$$",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przdXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcSTIe30CBuWmP1+uyGYZgZDHt0FLgpzbmBtaZEH6JjLtbA+zX4bD3u3gx65o2NJhMpxXNauH4WBlwhMiNwcYlXboN1Jhd8xIGBjH0OZzd+PbCI2sSeA74029n2yo277ZU="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=11a_{1}\\cos\\left(11t\\right)\\cdot\\:11"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=121a_{1}\\cos\\left(11t\\right)",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)-\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{^{\\prime}}+121\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)=11\\sin\\left(11t\\right)"
},
{
"type": "interim",
"title": "$$\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime}=a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)$$",
"input": "\\left(\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la suma/diferencia: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\left(a_{0}\\sin\\left(11t\\right)\\right)^{^{\\prime}}+\\left(a_{1}\\cos\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "$$\\left(a_{0}\\sin\\left(11t\\right)\\right)^{\\prime}=a_{0}\\cos\\left(11t\\right)\\cdot\\:11$$",
"input": "\\left(a_{0}\\sin\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=a_{0}\\left(\\sin\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\sin\\left(11t\\right)\\right)^{\\prime}",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\sin\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\sin\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"input": "\\left(\\sin\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\left(\\sin\\left(u\\right)\\right)^{\\prime}=\\cos\\left(u\\right)$$",
"result": "=\\cos\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przbRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19InMhwrDgcALtDbcjUqULjO0e5byrQDQVCXUD0vH/fvOdy0Sv9DakosoaSdvz3y7/jiRKGGP17LApxwbbz9SLndow=="
}
},
{
"type": "step",
"result": "=\\cos\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=11t$$",
"result": "=\\cos\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zq/dS8/GgjeXPl0DU/przdXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcSTIe30CBuWmP1+uyGYZgZDHt0FLgpzbmBtaZEH6JjLtbA+zX4bD3u3gx65o2NJhMpxXNauH4WBlwhMiNwcYlXboN1Jhd8xIGBjH0OZzd+PbCI2sSeA74029n2yo277ZU="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=a_{0}\\cos\\left(11t\\right)\\cdot\\:11"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "interim",
"title": "$$\\left(a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}=-11a_{1}\\sin\\left(11t\\right)$$",
"input": "\\left(a_{1}\\cos\\left(11t\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=a_{1}\\left(\\cos\\left(11t\\right)\\right)^{^{\\prime}}"
},
{
"type": "interim",
"title": "Aplicar la regla de la cadena:$${\\quad}-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}$$",
"input": "\\left(\\cos\\left(11t\\right)\\right)^{\\prime}",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de la cadena: $$\\frac{df\\left(u\\right)}{dx}=\\frac{df}{du}\\cdot\\frac{du}{dx}$$",
"secondary": [
"$$f=\\cos\\left(u\\right),\\:\\:u=11t$$"
],
"result": "=\\left(\\cos\\left(u\\right)\\right)^{^{\\prime}}\\left(11t\\right)^{^{\\prime}}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Chain%20Rule",
"practiceTopic": "Chain Rule"
}
},
{
"type": "interim",
"title": "$$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"input": "\\left(\\cos\\left(u\\right)\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$\\left(\\cos\\left(u\\right)\\right)^{\\prime}=-\\sin\\left(u\\right)$$",
"result": "=-\\sin\\left(u\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyLRGYgE/Uo1dnewG4GHneMqXIQHgliMhSOSNsNni19In4/yNG6RtaMeFUKXMVc75z2DpznShoZau34eGycg+7G3mVcoSp6dsHpd9bDcXuJ0pz2gvuHI/drUkZ7NmziNx4SS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "step",
"result": "=-\\sin\\left(u\\right)\\left(11t\\right)^{^{\\prime}}"
},
{
"type": "step",
"primary": "Sustituir en la ecuación $$u=11t$$",
"result": "=-\\sin\\left(11t\\right)\\left(11t\\right)^{^{\\prime}}"
}
],
"meta": {
"interimType": "Derivative Chain Rule 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7aJtZOJKzwzgX3sX8vWzwyNXWY62u60Y/NSU0TVZ6Sp6k3hxk9aCfAWodBRxXgUex4k6SVpXJ8ADj4hDb9X3WWeOBP9KtAhzuYXOSYJ7NErcOXl4x9OXOWFHQY9rAT64IGgK/z24D94dcZ1AqLbMwBPC30sSftAIFS6Qkpy19IkqbEmKQhIYyRwegIjh1uziVbUYzbseyJBZAzad0tEQA4ompXFf3SOUx+H18qfp3MLg="
}
},
{
"type": "interim",
"title": "$$\\left(11t\\right)^{\\prime}=11$$",
"input": "\\left(11t\\right)^{\\prime}",
"steps": [
{
"type": "step",
"primary": "Sacar la constante: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=11t^{^{\\prime}}"
},
{
"type": "step",
"primary": "Aplicar la regla de derivación: $$t^{\\prime}=1$$",
"result": "=11\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=11",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yOJE6/E13TW4tu/ku+cgCiENAk/2SHMUCwaiey+GXBFDkFJVC/dxv52FMorbXyXoHm2Q3+EpgfVSP40FbKzJvx8fRvP/NvzNmWlR8RepPuz+5lx9fdHWDvTjyCpqCxFp"
}
},
{
"type": "step",
"result": "=a_{1}\\left(-\\sin\\left(11t\\right)\\cdot\\:11\\right)"
},
{
"type": "step",
"primary": "Simplificar",
"result": "=-11a_{1}\\sin\\left(11t\\right)",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "=a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives"
}
},
{
"type": "step",
"result": "-121a_{0}\\sin\\left(11t\\right)-121a_{1}\\cos\\left(11t\\right)-\\left(a_{0}\\cos\\left(11t\\right)\\cdot\\:11-11a_{1}\\sin\\left(11t\\right)\\right)+121\\left(a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)\\right)=11\\sin\\left(11t\\right)"
},
{
"type": "step",
"primary": "Simplificar",
"result": "-11a_{0}\\cos\\left(11t\\right)+11a_{1}\\sin\\left(11t\\right)=11\\sin\\left(11t\\right)"
}
],
"meta": {
"interimType": "ODE Derive And Simplify 0Eq"
}
},
{
"type": "step",
"primary": "Encontrar una solución para el(los) coeficiente(s) $$a_{0},\\:a_{1}$$"
},
{
"type": "interim",
"title": "Resolver $$-11a_{0}\\cos\\left(11t\\right)+11a_{1}\\sin\\left(11t\\right)=11\\sin\\left(11t\\right):{\\quad}a_{0}=0,\\:a_{1}=1$$",
"steps": [
{
"type": "step",
"primary": "Agrupar términos semejantes",
"result": "-11a_{0}\\cos\\left(11t\\right)+11a_{1}\\sin\\left(11t\\right)=11\\sin\\left(11t\\right)"
},
{
"type": "step",
"primary": "Igualar los coeficientes de términos similares en ambos lados para crear una lista de ecuaciones",
"result": "\\begin{bmatrix}11=11a_{1}\\\\0=-11a_{0}\\end{bmatrix}"
},
{
"type": "interim",
"title": "Resolver sistema de ecuaciones:$${\\quad}a_{0}=0,\\:a_{1}=1$$",
"result": "a_{0}=0,\\:a_{1}=1",
"steps": [
{
"type": "step",
"result": "\\begin{bmatrix}11=11a_{1}\\\\0=-11a_{0}\\end{bmatrix}"
},
{
"type": "interim",
"title": "Despejar $$a_{1}\\:$$para $$11=11a_{1}:{\\quad}a_{1}=1$$",
"input": "11=11a_{1}",
"steps": [
{
"type": "step",
"primary": "Intercambiar lados",
"result": "11a_{1}=11"
},
{
"type": "interim",
"title": "Dividir ambos lados entre $$11$$",
"input": "11a_{1}=11",
"result": "a_{1}=1",
"steps": [
{
"type": "step",
"primary": "Dividir ambos lados entre $$11$$",
"result": "\\frac{11a_{1}}{11}=\\frac{11}{11}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "a_{1}=1"
}
],
"meta": {
"interimType": "Divide Both Sides Specific 1Eq",
"gptData": "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"
}
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Generic Isolate Title 2Eq"
}
},
{
"type": "step",
"result": "\\begin{bmatrix}0=-11a_{0}\\end{bmatrix}"
},
{
"type": "interim",
"title": "Despejar $$a_{0}\\:$$para $$0=-11a_{0}:{\\quad}a_{0}=0$$",
"input": "0=-11a_{0}",
"steps": [
{
"type": "step",
"primary": "Intercambiar lados",
"result": "-11a_{0}=0"
},
{
"type": "interim",
"title": "Dividir ambos lados entre $$-11$$",
"input": "-11a_{0}=0",
"result": "a_{0}=0",
"steps": [
{
"type": "step",
"primary": "Dividir ambos lados entre $$-11$$",
"result": "\\frac{-11a_{0}}{-11}=\\frac{0}{-11}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "a_{0}=0"
}
],
"meta": {
"interimType": "Divide Both Sides Specific 1Eq",
"gptData": "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"
}
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Generic Isolate Title 2Eq"
}
},
{
"type": "step",
"primary": "Las soluciones para el sistema de ecuaciones son:",
"result": "a_{0}=0,\\:a_{1}=1"
}
],
"meta": {
"solvingClass": "System of Equations",
"interimType": "Partial Fraction Solve System Equation 0Eq"
}
}
],
"meta": {
"interimType": "Generic Solve Title 1Eq"
}
},
{
"type": "step",
"primary": "Sustituir las soluciones de parametro en $$y=a_{0}\\sin\\left(11t\\right)+a_{1}\\cos\\left(11t\\right)$$",
"result": "y=0\\cdot\\:\\sin\\left(11t\\right)+1\\cdot\\:\\cos\\left(11t\\right)"
},
{
"type": "step",
"primary": "Simplificar",
"result": "y=\\cos\\left(11t\\right)"
},
{
"type": "step",
"primary": "Una solución particular $$y_{p}$$ para$${\\quad}y^{\\prime\\prime}\\left(t\\right)-y^{\\prime}\\left(t\\right)+121y=11\\sin\\left(11t\\right){\\quad}$$es:",
"result": "y=\\cos\\left(11t\\right)"
}
],
"meta": {
"interimType": "Generic Find That Satisfies Title 2Eq"
}
},
{
"type": "step",
"primary": "La solución general $$y=y_h+y_p$$ es:",
"result": "y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)+\\cos\\left(11t\\right)"
}
],
"meta": {
"interimType": "ODE Solve Linear 0Eq"
}
},
{
"type": "step",
"result": "y=e^{\\frac{t}{2}}\\left(c_{1}\\cos\\left(\\frac{\\sqrt{483}t}{2}\\right)+c_{2}\\sin\\left(\\frac{\\sqrt{483}t}{2}\\right)\\right)+\\cos\\left(11t\\right)"
}
],
"meta": {
"solvingClass": "ODE"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "t",
"plotRequest": "#>#ODE#>#y=e^{\\frac{t}{2}}(c_{1}\\cos(\\frac{\\sqrt{483}t}{2})+c_{2}\\sin(\\frac{\\sqrt{483}t}{2}))+\\cos(11t)"
}
}
},
"meta": {
"showVerify": true
}
}
Solución
Solución
Pasos de solución
Resolver EDO lineales: