What is the solution for y^{''}-y^'-12y=0,y(0)=1,y^'(0)=-2 ?

The solution for y^{''}-y^'-12y=0,y(0)=1,y^'(0)=-2 is y=(1-6/7)e^{4t}+6/7 e^{-3t}

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y^{''}-y^'-12y=0,y(0)=1,y^'(0)=-2

y^{^{''}} - y^{^{'}} - 12 y = 0, y (0) = 1, y^{^{'}} (0) = - 2

y = (1 - \frac{6}{7}) e^{4 t} + \frac{6}{7} e^{- 3 t}

Solution steps

y^{''} (t) - y^{'} (t) - 12 y = 0

Solve linear ODE:

y = (1 - \frac{6}{7}) e^{4 t} + \frac{6}{7} e^{- 3 t}

y = (1 - \frac{6}{7}) e^{4 t} + \frac{6}{7} e^{- 3 t}

\frac{d ^{2} y}{d x ^{2}} - 5 \frac{d y}{d x} + 4 y = 0

y^{^{'''}} + 7 y^{^{''}} + 15 y^{^{'}} + 9 y = 0

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

\frac{d ^{3} y}{d x ^{3}} - 3 \frac{d ^{2} y}{d x ^{2}} + 4 y = 0

\frac{d ^{2} y}{d x ^{2}} - 5 y = 0, y (0) = - 2, y (2) = 10

What is the solution for y^{''}-y^'-12y=0,y(0)=1,y^'(0)=-2 ?
The solution for y^{''}-y^'-12y=0,y(0)=1,y^'(0)=-2 is y=(1-6/7)e^{4t}+6/7 e^{-3t}