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

The solution for y^{''}+ky^'-2k^2y=0 is y=c_{1}e^{kt}+c_{2}e^{-2kt}

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} - 2 y^{^{'}} + y = 0, y (0) = 9, y (1) = 5

y^{^{''}} + 10 y^{^{'}} + 25 y = 0, y (0) = 2, y^{^{'}} (0) = - 13

\frac{3}{4} \frac{d ^{2} x}{d t ^{2}} + 72 x = 0, x (0) = \frac{1}{4}

y^{^{'''}} - 11 y^{^{''}} + 28 y^{^{'}} = 0

(d^{4} + 2 d^{2} + 1) y = 0

What is the solution for y^{''}+ky^'-2k^2y=0 ?
The solution for y^{''}+ky^'-2k^2y=0 is y=c_{1}e^{kt}+c_{2}e^{-2kt}