What is the solution for b*y=(a-v^2)y^{''} ?

The solution for b*y=(a-v^2)y^{''} is y=c_{1}e^{sqrt(b/(a-v^2))t}+c_{2}e^{-sqrt(b/(a-v^2))t}

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b*y=(a-v^2)y^{''}

b \cdot y = (a - v^{2}) y^{^{''}}

y = c_{1} e^{\frac{b}{a - v ^{2}} t} + c_{2} e^{- \frac{b}{a - v ^{2}} t}

Solution steps

b \cdot y = (a - v^{2}) y^{''} (t)

Solve linear ODE:

y = c_{1} e^{\frac{b}{a - v ^{2}} t} + c_{2} e^{- \frac{b}{a - v ^{2}} t}

y = c_{1} e^{\frac{b}{a - v ^{2}} t} + c_{2} e^{- \frac{b}{a - v ^{2}} t}

y^{^{''}} = 9 y^{^{'}} - 20 y

y^{^{''}} - 4 y^{^{''}} - 5 y = 0

0.25 x^{^{''}} + 2 x^{^{'}} + 2 x = 0

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

\frac{d ^{2} x}{d t ^{2}} + 12 \frac{d x}{d t} + 32 x = 0

What is the solution for b*y=(a-v^2)y^{''} ?
The solution for b*y=(a-v^2)y^{''} is y=c_{1}e^{sqrt(b/(a-v^2))t}+c_{2}e^{-sqrt(b/(a-v^2))t}