y^{''}+49y=t^2cos(7t)+te^{(-7t)}

Lösung

y^{^{''}} + 49 y = t^{2} cos (7 t) + t e^{(- 7 t)}

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

Schritte zur Lösung

y^{''} (t) + 49 y = t^{2} cos (7 t) + t e^{(- 7 t)}

Solve linear ODE:

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

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

y^{^{''}} + 4 y = e^{t}, y (0) = 0, y^{^{'}} (0) = 0

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

\frac{d ^{2} y}{d t ^{2}} + 2 \frac{d y}{d t} - 3 y = e^{t}

y^{^{''}} + 8 y^{^{'}} + 17 y = 6 e^{- 40} cos (t)

y^{^{''}} + \frac{3}{2} y^{^{'}} = x + e^{2 x}