What is the solution for -a^2y^{''}=ky ?

The solution for -a^2y^{''}=ky is y=c_{1}cos((sqrt(k)t)/a)+c_{2}sin((sqrt(k)t)/a)

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-a^2y^{''}=ky

- a^{2} y^{^{''}} = k y

y = c_{1} cos (\frac{k t}{a}) + c_{2} sin (\frac{k t}{a})

Solution steps

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

Solve linear ODE:

y = c_{1} cos (\frac{k t}{a}) + c_{2} sin (\frac{k t}{a})

y = c_{1} cos (\frac{k t}{a}) + c_{2} sin (\frac{k t}{a})

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

y^{^{''}} - 20 y^{^{'}} + 116 y = 0

9 y^{^{''}} - 6 y^{^{'}} + y = 0, y (0) = 1, y^{^{'}} (0) = 2

2 y^{^{''}} + 12 y^{^{'}} + 19 y = 0, y (0) = 2, y^{^{'}} (0) = 0

y^{^{''}} + 32 y^{^{'}} + 160 y = 0

What is the solution for -a^2y^{''}=ky ?
The solution for -a^2y^{''}=ky is y=c_{1}cos((sqrt(k)t)/a)+c_{2}sin((sqrt(k)t)/a)