What is the solution for (d^2y)/(dx^2)-6(dy)/(dx)+9y=(x+1)e^x ?

The solution for (d^2y)/(dx^2)-6(dy)/(dx)+9y=(x+1)e^x is y=c_{1}e^{3x}+c_{2}xe^{3x}+(e^xx)/4+1/2 e^x

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(d^2y)/(dx^2)-6(dy)/(dx)+9y=(x+1)e^x

\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 9 y = (x + 1) e^{x}

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

Solution steps

\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 9 y = (x + 1) e^{x}

Solve linear ODE:

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

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

y^{^{''}} - 4 y^{^{'}} + 2 y = 5 e^{x}

y^{^{''}} + 2 y^{^{'}} = 2 x

y^{^{''}} + 4 y^{^{'}} + 4 y = e^{- 2 t} ln (t)

y^{^{''}} + 2 y^{^{'}} + y = e - t

y^{^{'''}} + 4 y^{^{'}} = 4 x^{2} + 8 x

What is the solution for (d^2y)/(dx^2)-6(dy)/(dx)+9y=(x+1)e^x ?
The solution for (d^2y)/(dx^2)-6(dy)/(dx)+9y=(x+1)e^x is y=c_{1}e^{3x}+c_{2}xe^{3x}+(e^xx)/4+1/2 e^x