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

The solution for y^{''}-9y=0,y(0)=1,y^'(0)=-1 is y=(1-2/3)e^{3t}+2/3 e^{-3t}

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

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

y = (1 - \frac{2}{3}) e^{3 t} + \frac{2}{3} e^{- 3 t}

Solution steps

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

Solve linear ODE:

y = (1 - \frac{2}{3}) e^{3 t} + \frac{2}{3} e^{- 3 t}

y = (1 - \frac{2}{3}) e^{3 t} + \frac{2}{3} e^{- 3 t}

\frac{d ^{2} y}{d t ^{2}} + 4 y = 0

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

- 2 y^{^{''}} + 5 y^{^{'}} + 10 y = 0

y^{^{''}} + 2 y^{^{'}} - 8 y = 0, y (0) = 2, y^{^{'}} (0) = 16

100 y^{^{''}} + 180 y^{^{'}} + 145 y = 0

What is the solution for y^{''}-9y=0,y(0)=1,y^'(0)=-1 ?
The solution for y^{''}-9y=0,y(0)=1,y^'(0)=-1 is y=(1-2/3)e^{3t}+2/3 e^{-3t}