(d^2+9)y=30sin^3(x)

解

(d^{2} + 9) y = 30 sin^{3} (x)

y = c_{1} cos (3 x) + c_{2} sin (3 x) - 10 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 10 sin^{3} (x) cos (3 x) d x

解答ステップ

(d^{2} + 9) y = 30 sin^{3} (x)

線形 ODE を解く:

y = c_{1} cos (3 x) + c_{2} sin (3 x) - 10 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 10 sin^{3} (x) cos (3 x) d x

y = c_{1} cos (3 x) + c_{2} sin (3 x) - 10 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 10 sin^{3} (x) cos (3 x) d x