What is the solution for 9y^{''}+6y^'+y=0,y(0)=4,y^'(0)=(-13)/3 ?

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

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9y^{''}+6y^'+y=0,y(0)=4,y^'(0)=(-13)/3

9 y^{^{''}} + 6 y^{^{'}} + y = 0, y (0) = 4, y^{^{'}} (0) = \frac{- 13}{3}

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

Solution steps

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

Solve linear ODE:

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

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

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

8 y^{^{''}} - 25 y^{^{'}} + 3 y = 0

25 y^{^{''}} - 100 y^{^{'}} + 109 y = 0, y (0) = 9, y^{^{'}} (0) = 8

256 y^{^{''''}} = 81 y

3 y^{^{''}} + 15 y = 0

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