y^{''}-4y^'+4y=x^2e^2x-e^2x

解

y^{^{''}} - 4 y^{^{'}} + 4 y = x^{2} e^{2} x - e^{2} x

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + \frac{e ^{2} x ^{3}}{4} + \frac{3 e ^{2} x ^{2}}{4} + \frac{7 e ^{2} x}{8} + \frac{e ^{2}}{2}

解答ステップ

y^{''} (x) - 4 y^{'} (x) + 4 y = x^{2} e^{2} x - e^{2} x

線形 ODE を解く:

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + \frac{e ^{2} x ^{3}}{4} + \frac{3 e ^{2} x ^{2}}{4} + \frac{7 e ^{2} x}{8} + \frac{e ^{2}}{2}

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + \frac{e ^{2} x ^{3}}{4} + \frac{3 e ^{2} x ^{2}}{4} + \frac{7 e ^{2} x}{8} + \frac{e ^{2}}{2}