y^{''}+y^'+y=e^{-t}+sin(2t)

Lösung

y^{^{''}} + y^{^{'}} + y = e^{- t} + sin (2 t)

y = e^{- \frac{t}{2}} (c_{1} cos (\frac{3 t}{2}) + c_{2} sin (\frac{3 t}{2})) + e^{- t} - \frac{3}{13} sin (2 t) - \frac{2}{13} cos (2 t)

Schritte zur Lösung

y^{''} (t) + y^{'} (t) + y = e^{- t} + sin (2 t)

Solve linear ODE:

y = e^{- \frac{t}{2}} (c_{1} cos (\frac{3 t}{2}) + c_{2} sin (\frac{3 t}{2})) + e^{- t} - \frac{3}{13} sin (2 t) - \frac{2}{13} cos (2 t)

y = e^{- \frac{t}{2}} (c_{1} cos (\frac{3 t}{2}) + c_{2} sin (\frac{3 t}{2})) + e^{- t} - \frac{3}{13} sin (2 t) - \frac{2}{13} cos (2 t)

x^{^{''}} + 4 x = 32 sin (4 t)

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

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

y^{^{''}} + y = (sec (x))^{3}

y^{^{''}} + 3 y^{^{'}} = x