What is the solution for y^{''}+y^'-2y=2x,y(0)=0,y^'(0)=1 ?

The solution for y^{''}+y^'-2y=2x,y(0)=0,y^'(0)=1 is y=e^x-1/2 e^{-2x}-x-1/2

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y^{''}+y^'-2y=2x,y(0)=0,y^'(0)=1

y^{^{''}} + y^{^{'}} - 2 y = 2 x, y (0) = 0, y^{^{'}} (0) = 1

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 8 y^{^{'}} + 16 y = 1 - x + e^{4 x}

y^{^{''}} - y^{^{'}} - 6 y = csc (x)

y^{^{'''}} - 2 y^{^{''}} - 5 y^{^{'}} + 6 y = e^{- 2 x}

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

y^{^{'''}} + 3 y^{^{''}} + 9 y^{^{'}} + 27 y = e^{x} + 3

What is the solution for y^{''}+y^'-2y=2x,y(0)=0,y^'(0)=1 ?
The solution for y^{''}+y^'-2y=2x,y(0)=0,y^'(0)=1 is y=e^x-1/2 e^{-2x}-x-1/2