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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

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

y^{^{''}} + 16 y = 32 sec (2 x)

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

(d^{2} + 1) \cdot y = x \cdot cos (x)

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

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