What is the solution for y^{''}-5y^'-6y=12e^2t ?

The solution for y^{''}-5y^'-6y=12e^2t is y=c_{1}e^{6t}+c_{2}e^{-t}-2e^2t+(5e^2)/3

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y^{''}-5y^'-6y=12e^2t

y^{^{''}} - 5 y^{^{'}} - 6 y = 12 e^{2} t

y = c_{1} e^{6 t} + c_{2} e^{- t} - 2 e^{2} t + \frac{5 e ^{2}}{3}

Solution steps

y^{''} (t) - 5 y^{'} (t) - 6 y = 12 e^{2} t

Solve linear ODE:

y = c_{1} e^{6 t} + c_{2} e^{- t} - 2 e^{2} t + \frac{5 e ^{2}}{3}

y = c_{1} e^{6 t} + c_{2} e^{- t} - 2 e^{2} t + \frac{5 e ^{2}}{3}

\frac{\partial}{\partial x} (y e^{3 x})

\frac{d}{d x} (- 2 (2 x^{2} - 4 x - 1)^{2})

\frac{\partial}{\partial x} (2)

t \cdot y^{^{'}} + 4 y = 8 t

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

What is the solution for y^{''}-5y^'-6y=12e^2t ?
The solution for y^{''}-5y^'-6y=12e^2t is y=c_{1}e^{6t}+c_{2}e^{-t}-2e^2t+(5e^2)/3