y^{''}+2y^'+4y= x/(e^{2x)(1+x^2)}

Lösung

y^{^{''}} + 2 y^{^{'}} + 4 y = \frac{x}{e ^{2 x} ( 1 + x ^{2} )}

y = e^{- x} (c_{1} cos (3 x) + c_{2} sin (3 x)) + e^{- x} cos (3 x) \cdot \int - \frac{3 x sin ( 3 x )}{3 e ^{x} ( 1 + x ^{2} )} d x + e^{- x} sin (3 x) \cdot \int \frac{3 x cos ( 3 x )}{3 e ^{x} ( 1 + x ^{2} )} d x

Schritte zur Lösung

y^{''} (x) + 2 y^{'} (x) + 4 y = \frac{x}{e ^{2 x} ( 1 + x ^{2} )}

Solve linear ODE:

y = e^{- x} (c_{1} cos (3 x) + c_{2} sin (3 x)) + e^{- x} cos (3 x) \cdot \int - \frac{3 x sin ( 3 x )}{3 e ^{x} ( 1 + x ^{2} )} d x + e^{- x} sin (3 x) \cdot \int \frac{3 x cos ( 3 x )}{3 e ^{x} ( 1 + x ^{2} )} d x

y = e^{- x} (c_{1} cos (3 x) + c_{2} sin (3 x)) + e^{- x} cos (3 x) \cdot \int - \frac{3 x sin ( 3 x )}{3 e ^{x} ( 1 + x ^{2} )} d x + e^{- x} sin (3 x) \cdot \int \frac{3 x cos ( 3 x )}{3 e ^{x} ( 1 + x ^{2} )} d x

y^{^{''}} - y^{^{'}} + 64 y = 8 sin (8 t)

y^{^{''}} - 3 y^{^{'}} + 2 y = 2 x + 4

y^{^{''}} + 5 y^{^{'}} = x - 2 - 3 e^{2 x}

x^{^{''}} + 3 x^{^{'}} + 33 x = 8 cos (5 t)

x^{^{''}} + 16 x = cos (4 t)