What is the solution for y^{''}+2ky^'+k^4y=0 ?

The solution for y^{''}+2ky^'+k^4y=0 is y=c_{1}e^{k(sqrt(-k^2+1)-1)t}+c_{2}e^{-k(sqrt(-k^2+1)+1)t}

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y^{''}+2ky^'+k^4y=0

y^{^{''}} + 2 k y^{^{'}} + k^{4} y = 0

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

Solution steps

y^{''} (t) + 2 k y^{'} (t) + k^{4} y = 0

Solve linear ODE:

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

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

(9, 12), (- 2, - 17)

\int \frac{x ^{2}}{( x + 1 ) ( x + 2 ) ^{2}} d x

\int x^{2} e^{x y} d y

\int e^{\frac{- x}{3}} d x

\int_{- 2}^{3} \frac{27}{x ^{4}} d x

What is the solution for y^{''}+2ky^'+k^4y=0 ?
The solution for y^{''}+2ky^'+k^4y=0 is y=c_{1}e^{k(sqrt(-k^2+1)-1)t}+c_{2}e^{-k(sqrt(-k^2+1)+1)t}