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

The solution for y^{''}-y=2x^2-x-3 is y=c_{1}e^x+c_{2}e^{-x}-2x^2+x-1

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y^{''}-y=2x^2-x-3

y^{^{''}} - y = 2 x^{2} - x - 3

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

Solution steps

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

Solve linear ODE:

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

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

\frac{d ^{2} y}{d x ^{2}} - 16 \frac{d y}{d x} + 16 y = 16 (x - 1)

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

y^{^{''}} + 2 y^{^{'}} + 5 y = 8 e^{- t} cos (2 t)

y^{^{''}} + 4 y = 5 t^{2} e^{t}

y^{^{''}} + y = cos (2 t)

What is the solution for y^{''}-y=2x^2-x-3 ?
The solution for y^{''}-y=2x^2-x-3 is y=c_{1}e^x+c_{2}e^{-x}-2x^2+x-1