What is the solution for y^{''}-6y^'+9y=0,y(0)=0,y^'(0)=2 ?

The solution for y^{''}-6y^'+9y=0,y(0)=0,y^'(0)=2 is y=2e^{3t}t

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y^{''}-6y^'+9y=0,y(0)=0,y^'(0)=2

y^{^{''}} - 6 y^{^{'}} + 9 y = 0, y (0) = 0, y^{^{'}} (0) = 2

y = 2 e^{3 t} t

Solution steps

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

Solve linear ODE:

y = 2 e^{3 t} t

y = 2 e^{3 t} t

2 y^{^{'}} - y^{^{''}} = 0

y^{^{'''}} + 4 y^{^{''}} - 9 y^{^{'}} - 36 y = 0

\frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} + 6 y = 0

20 y^{^{''}} + 11 y^{^{'}} - 3 y = 0

49 y^{^{''}} + 112 y^{^{'}} + 64 y = 0

What is the solution for y^{''}-6y^'+9y=0,y(0)=0,y^'(0)=2 ?
The solution for y^{''}-6y^'+9y=0,y(0)=0,y^'(0)=2 is y=2e^{3t}t