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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

4 y^{^{''}} + 8 y^{^{'}} + 3 y = 0

\frac{5 s}{( s ^{2} + 4 ) \cdot ( s + 7 )}

f (x) = (\frac{5 x ^{2} + 6 x + 1}{3 x - 5})^{\frac{3}{2}}

\int - \frac{1}{2} x^{2} d x

\int \frac{6 x + 4}{3 x ^{2} + 4 x + 2} d x

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