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

Lösung

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

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + e^{2 x} \cdot \int - e^{- 2 x + 2} x^{2} ln (x) d x + \frac{e ^{2 + 2 x} x ( - 2 e ^{- 2 x} x ln ( x ) - e ^{- 2 x} ln ( x ) - e ^{- 2 x} + Ei ( - 2 x ) )}{4}

Schritte zur Lösung

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

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + e^{2 x} \cdot \int - e^{- 2 x + 2} x^{2} ln (x) d x + \frac{e ^{2 + 2 x} x ( - 2 e ^{- 2 x} x ln ( x ) - e ^{- 2 x} ln ( x ) - e ^{- 2 x} + Ei ( - 2 x ) )}{4}

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + e^{2 x} \cdot \int - e^{- 2 x + 2} x^{2} ln (x) d x + \frac{e ^{2 + 2 x} x ( - 2 e ^{- 2 x} x ln ( x ) - e ^{- 2 x} ln ( x ) - e ^{- 2 x} + Ei ( - 2 x ) )}{4}

y^{^{''}} - 8 y^{^{'}} - 4 y - 2 = 10 x^{2} - 2 x e^{2}

y^{^{''}} - y^{^{'}} - 6 y = 22 cos (3 x)

9 y^{^{''}} + 6 y^{^{'}} + y = 2, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} - 4 y^{^{'}} + 4 y = t

y^{^{''''}} + 4 y^{^{''}} = sin (2 t) + t e^{3 t} + 4