What is the solution for y^{''}+y=4e^x+x^2-x ?

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

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y^{''}+y=4e^x+x^2-x

y^{^{''}} + y = 4 e^{x} + x^{2} - x

y = c_{1} cos (x) + c_{2} sin (x) + 2 e^{x} + x^{2} - x - 2

Solution steps

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

Solve linear ODE:

y = c_{1} cos (x) + c_{2} sin (x) + 2 e^{x} + x^{2} - x - 2

y = c_{1} cos (x) + c_{2} sin (x) + 2 e^{x} + x^{2} - x - 2

y^{^{''}} + 4 y^{^{'}} + 3 y = 1

y^{^{''}} + 4 y = 9 sin (t), y (0) = 1, y^{^{'}} (0) = - 1

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

9 y^{^{''}} - 16 y = sin (x)

y^{^{''}} - 2 y^{^{'}} + 5 y = 2 cos^{2} (2 x)

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