y^{''}+y^'-2y=xe^x+1/(e^{2x)}

Lösung

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

y = c_{1} e^{x} + c_{2} e^{- 2 x} + \frac{1}{3} e^{x} (\frac{x ^{2}}{2} - \frac{1}{3} e^{- 3 x}) - \frac{1}{3} e^{- 2 x} (\frac{1}{9} (3 e^{3 x} x - e^{3 x}) + x)

Schritte zur Lösung

y^{''} (x) + y^{'} (x) - 2 y = x e^{x} + \frac{1}{e ^{2 x}}

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{- 2 x} + \frac{1}{3} e^{x} (\frac{x ^{2}}{2} - \frac{1}{3} e^{- 3 x}) - \frac{1}{3} e^{- 2 x} (\frac{1}{9} (3 e^{3 x} x - e^{3 x}) + x)

y = c_{1} e^{x} + c_{2} e^{- 2 x} + \frac{1}{3} e^{x} (\frac{x ^{2}}{2} - \frac{1}{3} e^{- 3 x}) - \frac{1}{3} e^{- 2 x} (\frac{1}{9} (3 e^{3 x} x - e^{3 x}) + x)

y^{^{''}} - 2 y^{^{'}} + y = \frac{e ^{x}}{( 1 + x ) ( 2 + x )}

\frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} - 45 y = e^{5 x}

\frac{d ^{2} x}{d t ^{2}} + 36 x = 10 sin (4 t)

y^{^{''''}} - y = 1, y (0) = 0, y^{^{'}} (0) = 0, y^{^{''}} (0) = 0

y^{^{''}} - 4 y^{^{'}} = 5 e^{2 x} - 2 cos (x)