What is the solution for 81y^{''}+108y^'-13y=0,y(0)=1,y^'(0)=0 ?

The solution for 81y^{''}+108y^'-13y=0,y(0)=1,y^'(0)=0 is y=(1-1/14)e^{t/9}+1/14 e^{-(13t)/9}

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81y^{''}+108y^'-13y=0,y(0)=1,y^'(0)=0

81 y^{^{''}} + 108 y^{^{'}} - 13 y = 0, y (0) = 1, y^{^{'}} (0) = 0

y = (1 - \frac{1}{14}) e^{\frac{t}{9}} + \frac{1}{14} e^{- \frac{13 t}{9}}

Solution steps

81 y^{''} (t) + 108 y^{'} (t) - 13 y = 0

Solve linear ODE:

y = (1 - \frac{1}{14}) e^{\frac{t}{9}} + \frac{1}{14} e^{- \frac{13 t}{9}}

y = (1 - \frac{1}{14}) e^{\frac{t}{9}} + \frac{1}{14} e^{- \frac{13 t}{9}}

x^{^{''}} + 0.5 x^{^{'}} + 4 x = 0

x^{^{''}} - c^{2} x = 0

y^{^{''}} - 8 y^{^{'}} + 65 y = 0, y (0) = 1, y^{^{'}} (0) = 2

y^{^{''}} - 9 y^{^{'}} + 22 y = 0

y^{^{''}} + 6 y + 25 y = 0, y (0) = 1, y^{^{'}} (0) = - 1

What is the solution for 81y^{''}+108y^'-13y=0,y(0)=1,y^'(0)=0 ?
The solution for 81y^{''}+108y^'-13y=0,y(0)=1,y^'(0)=0 is y=(1-1/14)e^{t/9}+1/14 e^{-(13t)/9}