What is the solution for kx^{''}+ax=0 ?

The solution for kx^{''}+ax=0 is x=c_{1}cos(sqrt(a/k)t)+c_{2}sin(sqrt(a/k)t)

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kx^{''}+ax=0

k x^{^{''}} + a x = 0

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

Solution steps

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

Solve linear ODE:

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

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

\frac{d ^{2} x}{d t ^{2}} + a^{2} x = 0

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

y^{^{'''}} + 5 y^{^{''}} + y^{^{'}} - 75 y = 0, y (0) = e^{3 x}

y^{^{''}} - 4 y = 0, y (0) = 1, y^{^{'}} (0) = 2

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

What is the solution for kx^{''}+ax=0 ?
The solution for kx^{''}+ax=0 is x=c_{1}cos(sqrt(a/k)t)+c_{2}sin(sqrt(a/k)t)