What is the solution for y^{''}+k(1+k)y=0 ?

The solution for y^{''}+k(1+k)y=0 is y=c_{1}e^{sqrt(-k)sqrt(k+1)t}+c_{2}e^{-sqrt(-k)sqrt(k+1)t}

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y^{''}+k(1+k)y=0

y^{^{''}} + k (1 + k) y = 0

y = c_{1} e^{- k k + 1 t} + c_{2} e^{- - k k + 1 t}

Solution steps

y^{''} (t) + k (1 + k) y = 0

Solve linear ODE:

y = c_{1} e^{- k k + 1 t} + c_{2} e^{- - k k + 1 t}

y = c_{1} e^{- k k + 1 t} + c_{2} e^{- - k k + 1 t}

y^{^{''}} + y^{^{'}} - 30 y = 0, y (0) = 1, y^{^{'}} (0) = 38

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

y^{^{''}} + 4 y^{^{'}} + 4 y = 0, y (0) = 5, y^{^{'}} (0) = - 13

y^{^{''}} - 9 y^{^{'}} + 6 y = 0

16 y^{^{''}} + 522.67 y = 0

What is the solution for y^{''}+k(1+k)y=0 ?
The solution for y^{''}+k(1+k)y=0 is y=c_{1}e^{sqrt(-k)sqrt(k+1)t}+c_{2}e^{-sqrt(-k)sqrt(k+1)t}