x^{''}+6x^'+8x=0,x(0)=1,x^'(1)=2

Lösung

x^{^{''}} + 6 x^{^{'}} + 8 x = 0, x (0) = 1, x^{^{'}} (1) = 2

x = (1 - \frac{2 e ^{4} + 2 e ^{2}}{2 ( e ^{2} - 2 )}) e^{- 2 t} + \frac{2 e ^{4} + 2 e ^{2}}{2 ( e ^{2} - 2 )} e^{- 4 t}

Schritte zur Lösung

x^{''} (t) + 6 x^{'} (t) + 8 x = 0

Solve linear ODE:

x = (1 - \frac{2 e ^{4} + 2 e ^{2}}{2 ( e ^{2} - 2 )}) e^{- 2 t} + \frac{2 e ^{4} + 2 e ^{2}}{2 ( e ^{2} - 2 )} e^{- 4 t}

x = (1 - \frac{2 e ^{4} + 2 e ^{2}}{2 ( e ^{2} - 2 )}) e^{- 2 t} + \frac{2 e ^{4} + 2 e ^{2}}{2 ( e ^{2} - 2 )} e^{- 4 t}

\frac{d ^{2} x}{d t ^{2}} + 2 \frac{d x}{d t} + x = 0

y^{^{''}} + 8 y^{^{'}} + 32 y = 0, y^{0} = 1, y^{^{'}} 0 = - 4

\frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} + 8 y = 0

y^{^{''}} + y = 0, y (0) = 3, y (0) = 30

x^{^{''}} + 5 x^{^{'}} + 9 x = 0