What is the solution for y^{''}+y^'-6y=4e^t,y(0)=0,y^'(0)=0 ?

The solution for y^{''}+y^'-6y=4e^t,y(0)=0,y^'(0)=0 is y=(-1/5+1)e^{2t}+1/5 e^{-3t}-e^t

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

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

y = (- \frac{1}{5} + 1) e^{2 t} + \frac{1}{5} e^{- 3 t} - e^{t}

Solution steps

y^{''} (t) + y^{'} (t) - 6 y = 4 e^{t}

Solve linear ODE:

y = (- \frac{1}{5} + 1) e^{2 t} + \frac{1}{5} e^{- 3 t} - e^{t}

y = (- \frac{1}{5} + 1) e^{2 t} + \frac{1}{5} e^{- 3 t} - e^{t}

x + x^{- \frac{1}{3}}

\frac{\partial}{\partial x} (3 x^{2} y - 3 x + 3 y^{3})

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

(3 x^{2} - 3 x + 1) x^{3}

\int \frac{x ^{3}}{( 1 - x ^{4} ) ^{7}} d x

What is the solution for y^{''}+y^'-6y=4e^t,y(0)=0,y^'(0)=0 ?
The solution for y^{''}+y^'-6y=4e^t,y(0)=0,y^'(0)=0 is y=(-1/5+1)e^{2t}+1/5 e^{-3t}-e^t