y^{''}+by^'+cy=e^xsin(x)

Lösung

y^{^{''}} + b y^{^{'}} + cy = e^{x} sin (x)

y = c_{1} e^{\frac{x ( - b + b ^{2} - 4 c )}{2}} + c_{2} e^{\frac{x ( - b - b ^{2} - 4 c )}{2}} + \frac{e ^{x} sin ( x ) ( b + c )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b} - \frac{e ^{x} cos ( x ) ( 2 + b )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b}

Schritte zur Lösung

y^{''} (x) + b y^{'} (x) + cy = e^{x} sin (x)

Solve linear ODE:

y = c_{1} e^{\frac{x ( - b + b ^{2} - 4 c )}{2}} + c_{2} e^{\frac{x ( - b - b ^{2} - 4 c )}{2}} + \frac{e ^{x} sin ( x ) ( b + c )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b} - \frac{e ^{x} cos ( x ) ( 2 + b )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b}

y = c_{1} e^{\frac{x ( - b + b ^{2} - 4 c )}{2}} + c_{2} e^{\frac{x ( - b - b ^{2} - 4 c )}{2}} + \frac{e ^{x} sin ( x ) ( b + c )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b} - \frac{e ^{x} cos ( x ) ( 2 + b )}{c ^{2} + 4 + 2 b ^{2} + 4 b + 2 c b}

10 y^{^{''}} + 50 y^{^{'}} + 57.6 y = cos (x)

y^{^{''}} + y = 5 sin (4 x)

y^{^{''}} - 3 y^{^{'}} - 2 y = 9 e^{- x}

(D^{3} + D^{2} + 5 D + 5) y = 5 cos (2 x)

y^{^{''}} + 4 y = x (sin (2 x))