What is the solution for y^{''}+6y^'+9y=12e^t,y(0)=-1,y^'(0)=7 ?

The solution for y^{''}+6y^'+9y=12e^t,y(0)=-1,y^'(0)=7 is y=-7/4 e^{-3t}+e^{-3t}t+3/4 e^t

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y^{''}+6y^'+9y=12e^t,y(0)=-1,y^'(0)=7

y^{^{''}} + 6 y^{^{'}} + 9 y = 12 e^{t}, y (0) = - 1, y^{^{'}} (0) = 7

y = - \frac{7}{4} e^{- 3 t} + e^{- 3 t} t + \frac{3}{4} e^{t}

Solution steps

y^{''} (t) + 6 y^{'} (t) + 9 y = 12 e^{t}

Solve linear ODE:

y = - \frac{7}{4} e^{- 3 t} + e^{- 3 t} t + \frac{3}{4} e^{t}

y = - \frac{7}{4} e^{- 3 t} + e^{- 3 t} t + \frac{3}{4} e^{t}

y^{^{''}} + y^{^{'}} = 2 cos (x)

y^{^{''}} + y^{^{'}} - 2 y = 2 x + 3, y (0) = 1, y^{^{'}} (0) = - 4

y^{^{''}} + 5 y^{^{'}} + 4 y = 10 t e^{5 t}

\frac{d ^{2} y}{d x ^{2}} - 3 \frac{d y}{d x} + 2 = 0

y^{^{''}} + 2 y^{^{'}} + y = e^{- t} \cdot ln (t)

What is the solution for y^{''}+6y^'+9y=12e^t,y(0)=-1,y^'(0)=7 ?
The solution for y^{''}+6y^'+9y=12e^t,y(0)=-1,y^'(0)=7 is y=-7/4 e^{-3t}+e^{-3t}t+3/4 e^t