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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - k^{2} y = 0, y (0) = 0, y^{^{'}} (0) = 1

x^{^{'''}} + 5 x^{^{''}} - 25 x^{^{'}} - 125 x = 0

y^{^{''}} - 12 y^{^{'}} + 35 y = 0, y (0) = e^{k x}

\frac{d ^{3} y}{d x ^{3}} + 5 y = 0

y^{^{''}} + 5 y^{^{'}} - 14 y = 0, y^{^{'}} (0) = 56

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