(d^2y)/(dx^2)+4y=sin^2(x)

Lösung

\frac{d ^{2} y}{d x ^{2}} + 4 y = sin^{2} (x)

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

Schritte zur Lösung

\frac{d ^{2} y}{d x ^{2}} + 4 y = sin^{2} (x)

Solve linear ODE:

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

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

e^{x} y^{^{''}} + 2 e^{x} y^{^{'}} + y e^{x} = ln (x)

y^{^{''}} + 7 y^{^{'}} + 6 y = 24

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

(d^{2} + 6 d + 10) y = 3 x e^{- 3 x} - 2 e^{3 x} cos (x)

y^{^{''}} + 2 y^{^{'}} - 8 y = 6 e^{- 3 x} - e^{- x}