(d^2+2d+1)y=cos^2(x)

Lösung

(d^{2} + 2 d + 1) y = cos^{2} (x)

y = c_{1} e^{- x} + c_{2} x e^{- x} - \frac{1}{2} x - \frac{1}{2} x cos (2 x) + x cos^{2} (x) - \frac{3}{50} cos (2 x) + \frac{2}{25} sin (2 x) + \frac{1}{2}

Schritte zur Lösung

(d^{2} + 2 d + 1) y = cos^{2} (x)

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} x e^{- x} - \frac{1}{2} x - \frac{1}{2} x cos (2 x) + x cos^{2} (x) - \frac{3}{50} cos (2 x) + \frac{2}{25} sin (2 x) + \frac{1}{2}

y = c_{1} e^{- x} + c_{2} x e^{- x} - \frac{1}{2} x - \frac{1}{2} x cos (2 x) + x cos^{2} (x) - \frac{3}{50} cos (2 x) + \frac{2}{25} sin (2 x) + \frac{1}{2}

(d^{2} + 5 d + 6) y = e^{- 2} x sin^{2} (x)

so l v e f or y, (d^{2} + 4 d + 4) y = sin (t)

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

(d - 1)^{4} y = e^{x}

(d - 2)^{2} y = 8 (e^{2} x + sin^{2} (x) + x^{2})