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

The solution for y^{''}-2y^{''}+y=4e^x+4e^{-x} is y=c_{1}e^x+c_{2}e^{-x}-2xe^x+2xe^{-x}

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

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

y = c_{1} e^{x} + c_{2} e^{- x} - 2 x e^{x} + 2 x e^{- x}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{- x} - 2 x e^{x} + 2 x e^{- x}

y = c_{1} e^{x} + c_{2} e^{- x} - 2 x e^{x} + 2 x e^{- x}

2 y^{^{''}} + 4 y = 6 t, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} + 4 y = t^{2} + 6 e^{t}, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} + 4 y = t^{2} + 6 e^{t}, y (0) = 0, y^{^{'}} (0) = 9

x^{^{''}} + x = 9 cos (4 t)

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

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