What is the solution for y^{'''}-3y^'+2y=9e^x ?

The solution for y^{'''}-3y^'+2y=9e^x is y=c_{1}e^{-2x}+c_{2}e^x+c_{3}xe^x+(3e^xx^2)/2

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y^{'''}-3y^'+2y=9e^x

y^{^{'''}} - 3 y^{^{'}} + 2 y = 9 e^{x}

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

Solution steps

y^{'''} (x) - 3 y^{'} (x) + 2 y = 9 e^{x}

Solve linear ODE:

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

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

y^{^{''}} - 6 y^{^{'}} + 5 y = 3^{x}

(d^{2} - 4 d + 4) y = x^{2} + e^{x} + cos^{2} (x)

y^{^{''}} + 4 y^{^{'}} = sin^{3} (x)

(d - 1)^{2} (d^{2} + 1) y = 3 x^{2} - 5 x + 2

y^{^{''}} - 4 y = (x^{2} - 4) sin (2 x)

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