(D^2+9)y=cos^2(x)

Lösung

(D^{2} + 9) y = cos^{2} (x)

y = c_{1} cos (3 x) + c_{2} sin (3 x) - \frac{1}{6} cos (3 x) (- \frac{1}{3} cos (3 x) + \frac{1}{2} (- \frac{1}{5} cos (5 x) - cos (x))) + \frac{1}{3} sin (3 x) (\frac{1}{3} cos^{2} (x) sin (3 x) + \frac{1}{6} (sin (x) - \frac{1}{5} sin (5 x)))

Schritte zur Lösung

(D^{2} + 9) y = cos^{2} (x)

Solve linear ODE:

y = c_{1} cos (3 x) + c_{2} sin (3 x) - \frac{1}{6} cos (3 x) (- \frac{1}{3} cos (3 x) + \frac{1}{2} (- \frac{1}{5} cos (5 x) - cos (x))) + \frac{1}{3} sin (3 x) (\frac{1}{3} cos^{2} (x) sin (3 x) + \frac{1}{6} (sin (x) - \frac{1}{5} sin (5 x)))

y = c_{1} cos (3 x) + c_{2} sin (3 x) - \frac{1}{6} cos (3 x) (- \frac{1}{3} cos (3 x) + \frac{1}{2} (- \frac{1}{5} cos (5 x) - cos (x))) + \frac{1}{3} sin (3 x) (\frac{1}{3} cos^{2} (x) sin (3 x) + \frac{1}{6} (sin (x) - \frac{1}{5} sin (5 x)))

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

y^{^{''}} + 60 y^{^{'}} + 900 y = - 360 x e^{- 4 x}

y^{^{''}} - 8 y^{^{'}} + 16 y = e^{4} x

y^{^{''}} - y = 12 x^{2} + e^{x}

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