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

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

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(d^2y)/(dx^2)+6(dy)/(dx)+9y=xe^{-3x}

\frac{d ^{2} y}{d x ^{2}} + 6 \frac{d y}{d x} + 9 y = x e^{- 3 x}

y = c_{1} e^{- 3 x} + c_{2} x e^{- 3 x} + \frac{e ^{- 3 x} x ^{3}}{6}

Solution steps

\frac{d ^{2} y}{d x ^{2}} + 6 \frac{d y}{d x} + 9 y = x e^{- 3 x}

Solve linear ODE:

y = c_{1} e^{- 3 x} + c_{2} x e^{- 3 x} + \frac{e ^{- 3 x} x ^{3}}{6}

y = c_{1} e^{- 3 x} + c_{2} x e^{- 3 x} + \frac{e ^{- 3 x} x ^{3}}{6}

y^{^{''}} + 11 y^{^{'}} + 18 y = 8 H (t)

y^{^{''}} - y = t, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} - y = t, y (0) = 0, y^{^{'}} (0) = 1

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

y^{^{''}} + y^{^{'}} = 2 x + 3

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