What is the solution for y^{''}+6y^'+9y=0,y(0)=4,y^'(0)=1 ?

The solution for y^{''}+6y^'+9y=0,y(0)=4,y^'(0)=1 is y=4e^{-3t}+13te^{-3t}

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y^{''}+6y^'+9y=0,y(0)=4,y^'(0)=1

y^{^{''}} + 6 y^{^{'}} + 9 y = 0, y (0) = 4, y^{^{'}} (0) = 1

y = 4 e^{- 3 t} + 13 t e^{- 3 t}

Solution steps

y^{''} (t) + 6 y^{'} (t) + 9 y = 0

Solve linear ODE:

y = 4 e^{- 3 t} + 13 t e^{- 3 t}

y = 4 e^{- 3 t} + 13 t e^{- 3 t}

\frac{d ^{2} x}{d t ^{2}} - 2 \frac{d x}{d t} - 15 x = 0

\frac{d ^{2} y}{d t ^{2}} - 20 \frac{d y}{d t} + 100 y = 0

y^{^{''}} - 10 y^{^{'}} - 171 y = 0, y (0) = 9, y^{^{'}} (0) = 3

y^{^{''}} - 19 y^{^{'}} - 10 y = 0, y (0) = 1, y^{^{'}} (0) = - 1

y^{^{''}} + 9 y = 0, y (\frac{π}{2}) = 6, y^{^{'}} (0) = 0

What is the solution for y^{''}+6y^'+9y=0,y(0)=4,y^'(0)=1 ?
The solution for y^{''}+6y^'+9y=0,y(0)=4,y^'(0)=1 is y=4e^{-3t}+13te^{-3t}