y^{''}+49y=x^2cos(7x)+xe^{-7x}

Lösung

y^{^{''}} + 49 y = x^{2} cos (7 x) + x e^{- 7 x}

y = c_{1} cos (7 x) + c_{2} sin (7 x) + \frac{x ^{3} sin ( 7 x )}{42} + \frac{x ^{2} cos ( 7 x )}{196} - \frac{x sin ( 7 x )}{1372} + \frac{e ^{- 7 x} x}{98} + \frac{1}{686} e^{- 7 x}

Schritte zur Lösung

y^{''} (x) + 49 y = x^{2} cos (7 x) + x e^{- 7 x}

Solve linear ODE:

y = c_{1} cos (7 x) + c_{2} sin (7 x) + \frac{x ^{3} sin ( 7 x )}{42} + \frac{x ^{2} cos ( 7 x )}{196} - \frac{x sin ( 7 x )}{1372} + \frac{e ^{- 7 x} x}{98} + \frac{1}{686} e^{- 7 x}

y = c_{1} cos (7 x) + c_{2} sin (7 x) + \frac{x ^{3} sin ( 7 x )}{42} + \frac{x ^{2} cos ( 7 x )}{196} - \frac{x sin ( 7 x )}{1372} + \frac{e ^{- 7 x} x}{98} + \frac{1}{686} e^{- 7 x}

y^{^{''}} + 4 y^{^{'}} = cos (x)

y^{^{''}} + 7 y^{^{'}} + 10 y = - 10 + e^{4 t}

y^{^{''}} + 2 y^{^{'}} + 2 y = (e^{(- x)}) \cdot cos (x)

y^{^{''}} - 3 y^{^{'}} - 4 y = x e^{x} + 2 cos (x) - sin (x)

D (D - 1)^{3} y = (x^{2} + 2 x) e^{x}