What is the solution for y^{''}+2y^'+y=7+75sin(2x) ?

The solution for y^{''}+2y^'+y=7+75sin(2x) is y=c_{1}e^{-x}+c_{2}xe^{-x}+7-9sin(2x)-12cos(2x)

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y^{''}+2y^'+y=7+75sin(2x)

y^{^{''}} + 2 y^{^{'}} + y = 7 + 75 sin (2 x)

y = c_{1} e^{- x} + c_{2} x e^{- x} + 7 - 9 sin (2 x) - 12 cos (2 x)

Solution steps

y^{''} (x) + 2 y^{'} (x) + y = 7 + 75 sin (2 x)

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} x e^{- x} + 7 - 9 sin (2 x) - 12 cos (2 x)

y = c_{1} e^{- x} + c_{2} x e^{- x} + 7 - 9 sin (2 x) - 12 cos (2 x)

y^{^{''}} + 2 y^{^{'}} - 3 y = (- 4 x^{2} - 4 x + 2) e^{2 x}

y^{^{''''}} + 2 y^{^{''}} + y = 4 + cos (2 t)

y^{^{''}} + 9 y = 4, y (0) = 0, y^{^{'}} (0) = 0

2 \frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} - 2 y = x e^{- 2 x}

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

What is the solution for y^{''}+2y^'+y=7+75sin(2x) ?
The solution for y^{''}+2y^'+y=7+75sin(2x) is y=c_{1}e^{-x}+c_{2}xe^{-x}+7-9sin(2x)-12cos(2x)