What is the solution for y^{''}-y^'-2y=4x^2,y(0)=1,y^'(0)=4 ?

The solution for y^{''}-y^'-2y=4x^2,y(0)=1,y^'(0)=4 is y=2e^{2x}+2e^{-x}-2x^2+2x-3

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y^{''}-y^'-2y=4x^2,y(0)=1,y^'(0)=4

y^{^{''}} - y^{^{'}} - 2 y = 4 x^{2}, y (0) = 1, y^{^{'}} (0) = 4

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

Solution steps

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

Solve linear ODE:

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

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

What is the solution for y^{''}-y^'-2y=4x^2,y(0)=1,y^'(0)=4 ?
The solution for y^{''}-y^'-2y=4x^2,y(0)=1,y^'(0)=4 is y=2e^{2x}+2e^{-x}-2x^2+2x-3