y^{''}+ay^'+by=t+e^{3t}

Lösung

y^{^{''}} + a y^{^{'}} + b y = t + e^{3 t}

y = c_{1} e^{\frac{t ( - a + a ^{2} - 4 b )}{2}} + c_{2} e^{\frac{t ( - a - a ^{2} - 4 b )}{2}} + \frac{t}{b} - \frac{a}{b ^{2}} + \frac{e ^{3 t}}{9 + 3 a + b}

Schritte zur Lösung

y^{''} (t) + a y^{'} (t) + b y = t + e^{3 t}

Solve linear ODE:

y = c_{1} e^{\frac{t ( - a + a ^{2} - 4 b )}{2}} + c_{2} e^{\frac{t ( - a - a ^{2} - 4 b )}{2}} + \frac{t}{b} - \frac{a}{b ^{2}} + \frac{e ^{3 t}}{9 + 3 a + b}

y = c_{1} e^{\frac{t ( - a + a ^{2} - 4 b )}{2}} + c_{2} e^{\frac{t ( - a - a ^{2} - 4 b )}{2}} + \frac{t}{b} - \frac{a}{b ^{2}} + \frac{e ^{3 t}}{9 + 3 a + b}

x \to - 1 lim (ln (x + 1))

\frac{\partial}{\partial x} (\frac{1}{2} (x^{2} + 4 y^{2}))

\frac{\partial}{\partial x} (\frac{1}{2 x})

2 x y \frac{d y}{d x} = y^{2} - x^{2} + 1

t an g e n t 100 x^{2} + 400 x + 5000, 0.5