What is the solution for 9y^{''}-y=xe^{x/3},y(0)=1,y(0)=0 ?

The solution for 9y^{''}-y=xe^{x/3},y(0)=1,y(0)=0 is

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9y^{''}-y=xe^{x/3},y(0)=1,y(0)=0

9 y^{^{''}} - y = x e^{\frac{x}{3}}, y (0) = 1, y (0) = 0

y = (- c_{2} + \frac{5}{8}) e^{\frac{x}{3}} + c_{2} e^{- \frac{x}{3}} + \frac{e ^{\frac{x}{3}} x ^{2}}{12} - \frac{1}{8} e^{- \frac{x}{3}} (2 e^{\frac{2 x}{3}} x - 3 e^{\frac{2 x}{3}})

Solution steps

9 y^{''} (x) - y = x e^{\frac{x}{3}}

Solve linear ODE:

y = (- c_{2} + \frac{5}{8}) e^{\frac{x}{3}} + c_{2} e^{- \frac{x}{3}} + \frac{e ^{\frac{x}{3}} x ^{2}}{12} - \frac{1}{8} e^{- \frac{x}{3}} (2 e^{\frac{2 x}{3}} x - 3 e^{\frac{2 x}{3}})

y = (- c_{2} + \frac{5}{8}) e^{\frac{x}{3}} + c_{2} e^{- \frac{x}{3}} + \frac{e ^{\frac{x}{3}} x ^{2}}{12} - \frac{1}{8} e^{- \frac{x}{3}} (2 e^{\frac{2 x}{3}} x - 3 e^{\frac{2 x}{3}})

What is the solution for 9y^{''}-y=xe^{x/3},y(0)=1,y(0)=0 ?
The solution for 9y^{''}-y=xe^{x/3},y(0)=1,y(0)=0 is