What is the solution for y^{''}+18y^'+81y=x,y(0)=0,y^'(0)=0 ?

The solution for y^{''}+18y^'+81y=x,y(0)=0,y^'(0)=0 is y= 2/729 e^{-9x}+(e^{-9x}x)/(81)+x/(81)-2/729

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y^{''}+18y^'+81y=x,y(0)=0,y^'(0)=0

y^{^{''}} + 18 y^{^{'}} + 81 y = x, y (0) = 0, y^{^{'}} (0) = 0

y = \frac{2}{729} e^{- 9 x} + \frac{e ^{- 9 x} x}{81} + \frac{x}{81} - \frac{2}{729}

Solution steps

y^{''} (x) + 18 y^{'} (x) + 81 y = x

Solve linear ODE:

y = \frac{2}{729} e^{- 9 x} + \frac{e ^{- 9 x} x}{81} + \frac{x}{81} - \frac{2}{729}

y = \frac{2}{729} e^{- 9 x} + \frac{e ^{- 9 x} x}{81} + \frac{x}{81} - \frac{2}{729}

What is the solution for y^{''}+18y^'+81y=x,y(0)=0,y^'(0)=0 ?
The solution for y^{''}+18y^'+81y=x,y(0)=0,y^'(0)=0 is y= 2/729 e^{-9x}+(e^{-9x}x)/(81)+x/(81)-2/729