(d^2+9)y=sin^3(x)

Lösung

(d^{2} + 9) y = sin^{3} (x)

y = c_{1} cos (3 x) + c_{2} sin (3 x) - \frac{1}{3} cos (3 x) (- sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x)) + sin (3 x) \cdot \int \frac{sin ^{3} ( x ) cos ( 3 x )}{3} d x

Schritte zur Lösung

(d^{2} + 9) y = sin^{3} (x)

Solve linear ODE:

y = c_{1} cos (3 x) + c_{2} sin (3 x) - \frac{1}{3} cos (3 x) (- sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x)) + sin (3 x) \cdot \int \frac{sin ^{3} ( x ) cos ( 3 x )}{3} d x

y = c_{1} cos (3 x) + c_{2} sin (3 x) - \frac{1}{3} cos (3 x) (- sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x)) + sin (3 x) \cdot \int \frac{sin ^{3} ( x ) cos ( 3 x )}{3} d x

[d^{2} - d - 12] \cdot y = e^{3 x}

(d) h (x) = \frac{1}{( 1 + x ^{2} )}

(d^{2} + 3 d + 1) \cdot y = x^{2} \cdot e^{3 x}

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

(d + d + d^{2}) y = 5 x^{2}