What is the solution for (d^2y)/(dx^2)+2(dy)/(dx)+5y=0 ?

The solution for (d^2y)/(dx^2)+2(dy)/(dx)+5y=0 is y=e^{-x}(c_{1}cos(2x)+c_{2}sin(2x))

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(d^2y)/(dx^2)+2(dy)/(dx)+5y=0

\frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} + 5 y = 0

y = e^{- x} (c_{1} cos (2 x) + c_{2} sin (2 x))

Solution steps

\frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} + 5 y = 0

Solve linear ODE:

y = e^{- x} (c_{1} cos (2 x) + c_{2} sin (2 x))

y = e^{- x} (c_{1} cos (2 x) + c_{2} sin (2 x))

y^{^{''}} + 22 y^{^{'}} + 122 y = 0, y (0) = 1, y^{^{'}} (0) = 0

4 y^{^{''}} + 20 y^{^{'}} + 169 y = 0

y^{^{''}} - 4 y^{^{'}} + 13 y = 0, y (0) = - 5, y^{^{'}} (0) = - 1

- y^{^{''}} + y = 0

y^{^{'''}} = y

What is the solution for (d^2y)/(dx^2)+2(dy)/(dx)+5y=0 ?
The solution for (d^2y)/(dx^2)+2(dy)/(dx)+5y=0 is y=e^{-x}(c_{1}cos(2x)+c_{2}sin(2x))