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

The solution for y^{''}+9y^'=e^t,y(0)=0,y^'(0)=0 is y=-1/90-1/10+1/90 e^{-9t}+1/10 e^t

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y^{''}+9y^'=e^t,y(0)=0,y^'(0)=0

y^{^{''}} + 9 y^{^{'}} = e^{t}, y (0) = 0, y^{^{'}} (0) = 0

y = - \frac{1}{90} - \frac{1}{10} + \frac{1}{90} e^{- 9 t} + \frac{1}{10} e^{t}

Solution steps

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

Solve linear ODE:

y = - \frac{1}{90} - \frac{1}{10} + \frac{1}{90} e^{- 9 t} + \frac{1}{10} e^{t}

y = - \frac{1}{90} - \frac{1}{10} + \frac{1}{90} e^{- 9 t} + \frac{1}{10} e^{t}

y^{^{''}} - 4 y^{^{'}} + 4 y = 2 x - sin (2 x)

y^{^{''}} - y = 3 x e^{x} cos (2 x)

y^{^{''}} + 4 y^{^{'}} + 5 y = 5

4 y^{^{''}} + 4 y^{^{'}} + y = \frac{5 e ^{- \frac{t}{2}}}{t ^{2} + 1}

y^{^{''}} - 5 y^{^{'}} + 6 y = - 3 sin (2 x)

What is the solution for y^{''}+9y^'=e^t,y(0)=0,y^'(0)=0 ?
The solution for y^{''}+9y^'=e^t,y(0)=0,y^'(0)=0 is y=-1/90-1/10+1/90 e^{-9t}+1/10 e^t