(D^2-2D+3)y=x^3+sin(x)

Lösung

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

y = e^{x} (c_{1} cos (2 x) + c_{2} sin (2 x)) + \frac{x ^{3} + 2 x ^{2}}{3} + \frac{2 x}{9} - \frac{8}{27} + \frac{1}{4} sin (x) + \frac{1}{4} cos (x)

Schritte zur Lösung

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

Solve linear ODE:

y = e^{x} (c_{1} cos (2 x) + c_{2} sin (2 x)) + \frac{x ^{3} + 2 x ^{2}}{3} + \frac{2 x}{9} - \frac{8}{27} + \frac{1}{4} sin (x) + \frac{1}{4} cos (x)

y = e^{x} (c_{1} cos (2 x) + c_{2} sin (2 x)) + \frac{x ^{3} + 2 x ^{2}}{3} + \frac{2 x}{9} - \frac{8}{27} + \frac{1}{4} sin (x) + \frac{1}{4} cos (x)

y^{^{''}} + y = 16 cos (t), y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} + 9 y = 2 x - 5

y^{^{''}} - y^{^{'}} + y = 5 sin (t)

x^{^{''}} + 16 x = 144, x (0) = 5, x^{^{'}} (0) = 12

x^{^{''}} + 6 x^{^{'}} + 9 = 0