What is the solution for y^{''}-y=xsin(x) ?

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

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y^{''}-y=xsin(x)

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 4 y = - 12 sin (2 x)

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

y^{^{'''}} - y^{^{''}} + y^{^{'}} - y = x e^{x} - e^{- x} + 7

y^{^{''}} + 36 y = csc (6 x)

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

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