What is the solution for 2y^{''}+y^'+1=2e^2x ?

The solution for 2y^{''}+y^'+1=2e^2x is y=c_{1}+c_{2}e^{-x/2}+e^2x^2+(-1-4e^2)x

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2y^{''}+y^'+1=2e^2x

2 y^{^{''}} + y^{^{'}} + 1 = 2 e^{2} x

y = c_{1} + c_{2} e^{- \frac{x}{2}} + e^{2} x^{2} + (- 1 - 4 e^{2}) x

Solution steps

2 y^{''} (x) + y^{'} (x) + 1 = 2 e^{2} x

Solve linear ODE:

y = c_{1} + c_{2} e^{- \frac{x}{2}} + e^{2} x^{2} + (- 1 - 4 e^{2}) x

y = c_{1} + c_{2} e^{- \frac{x}{2}} + e^{2} x^{2} + (- 1 - 4 e^{2}) x

What is the solution for 2y^{''}+y^'+1=2e^2x ?
The solution for 2y^{''}+y^'+1=2e^2x is y=c_{1}+c_{2}e^{-x/2}+e^2x^2+(-1-4e^2)x