What is the solution for y^{''}+a*y=0 ?

The solution for y^{''}+a*y=0 is y=c_{1}cos(sqrt(a)t)+c_{2}sin(sqrt(a)t)

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y^{''}+a*y=0

y^{^{''}} + a \cdot y = 0

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

Solution steps

y^{''} (t) + a \cdot y = 0

Solve linear ODE:

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

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

y^{^{''}} - 4 c^{2} y = 0

\frac{d ^{2} y}{d t ^{2}} + 5 \frac{d y}{d t} + 4 y = 0

y^{^{'''}} - 2 y^{^{''}} - 11 y^{^{'}} + 12 y = 0

(y^{^{'''''}} + 8 y^{^{'''}} + 16 y^{^{'}}) y = 0

y^{^{'}} = 6 y^{^{''}}

What is the solution for y^{''}+a*y=0 ?
The solution for y^{''}+a*y=0 is y=c_{1}cos(sqrt(a)t)+c_{2}sin(sqrt(a)t)