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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + y^{^{'}} + y = 0, y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} - 6 y^{^{'}} + 9 y = 0, y (0) = 13, y (1) = 0

x^{^{''}} + 6 x^{^{'}} + 8 x = 0

4 y^{^{''}} - 4 y^{^{'}} + 3 y = 0, y (0) = 1, y^{^{'}} (0) = 5

y^{^{''}} + 2 y^{^{'}} + 10 y = 0, y (\frac{π}{2}) = 0, y^{^{'}} (\frac{π}{2}) = 6

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