What is the solution for y^{''}+y^'=4x+10sin(x) ?

The solution for y^{''}+y^'=4x+10sin(x) is y=c_{1}+c_{2}e^{-x}+2x^2-4x-5cos(x)-5sin(x)+4

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y^{''}+y^'=4x+10sin(x)

y^{^{''}} + y^{^{'}} = 4 x + 10 sin (x)

y = c_{1} + c_{2} e^{- x} + 2 x^{2} - 4 x - 5 cos (x) - 5 sin (x) + 4

Solution steps

y^{''} (x) + y^{'} (x) = 4 x + 10 sin (x)

Solve linear ODE:

y = c_{1} + c_{2} e^{- x} + 2 x^{2} - 4 x - 5 cos (x) - 5 sin (x) + 4

y = c_{1} + c_{2} e^{- x} + 2 x^{2} - 4 x - 5 cos (x) - 5 sin (x) + 4

y^{^{''}} - 4 y^{^{'}} + 3 y = e t - 2 t + 3

y^{^{''}} - 3 y^{^{'}} = 2 x + 5 - e^{3 x}

y^{^{''}} - y^{^{'}} - 6 y = 2 e^{3 t}

y^{^{'''}} - 3 y^{^{'}} - 2 y = 9 e^{- x}

y^{^{''}} + y = 2 sin (2 t), y (0) = 11, y^{^{'}} (0) = 0

What is the solution for y^{''}+y^'=4x+10sin(x) ?
The solution for y^{''}+y^'=4x+10sin(x) is y=c_{1}+c_{2}e^{-x}+2x^2-4x-5cos(x)-5sin(x)+4