What is the solution for y^{''}+2*y^'+y=6*x*e^{-x} ?

The solution for y^{''}+2*y^'+y=6*x*e^{-x} is y=c_{1}e^{-x}+c_{2}xe^{-x}+e^{-x}x^3

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y^{''}+2y^'+y=6x*e^{-x}

y^{^{''}} + 2 \cdot y^{^{'}} + y = 6 \cdot x \cdot e^{- x}

y = c_{1} e^{- x} + c_{2} x e^{- x} + e^{- x} x^{3}

Solution steps

y^{''} (x) + 2 \cdot y^{'} (x) + y = 6 \cdot x \cdot e^{- x}

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} x e^{- x} + e^{- x} x^{3}

y = c_{1} e^{- x} + c_{2} x e^{- x} + e^{- x} x^{3}

\int tan^{2} (4 x) sec^{4} (4 x) d x

2 x (1.2)

\int sin^{5} (12 x) d x

\int_{1}^{3} \frac{1}{x ( 1 + x )} d x

x \to 0 + lim (x^{x^{2}})

What is the solution for y^{''}+2*y^'+y=6*x*e^{-x} ?
The solution for y^{''}+2*y^'+y=6*x*e^{-x} is y=c_{1}e^{-x}+c_{2}xe^{-x}+e^{-x}x^3