y^{''}-4y^'+4y=xsin(x)

解

y^{^{''}} - 4 y^{^{'}} + 4 y = x sin (x)

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + \frac{3 x sin ( x )}{25} + \frac{4 x cos ( x )}{25} + \frac{4}{125} sin (x) + \frac{22}{125} cos (x)

解答ステップ

y^{''} (x) - 4 y^{'} (x) + 4 y = x sin (x)

線形 ODE を解く:

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + \frac{3 x sin ( x )}{25} + \frac{4 x cos ( x )}{25} + \frac{4}{125} sin (x) + \frac{22}{125} cos (x)

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + \frac{3 x sin ( x )}{25} + \frac{4 x cos ( x )}{25} + \frac{4}{125} sin (x) + \frac{22}{125} cos (x)