What is the solution for y^{''}+a^2*n^2*pi^2*y=0 ?

The solution for y^{''}+a^2*n^2*pi^2*y=0 is y=c_{1}cos(piant)+c_{2}sin(piant)

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y^{''}+a^2n^2pi^2*y=0

y^{^{''}} + a^{2} \cdot n^{2} \cdot π^{2} \cdot y = 0

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

Solution steps

y^{''} (t) + a^{2} \cdot n^{2} \cdot π^{2} \cdot y = 0

Solve linear ODE:

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

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

3 y^{^{''}} - 4 y^{^{'}} = 0

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} + 29 y = 0

y^{^{'''}} - 5 y^{^{''}} + 3 y^{^{'}} + y = 0

64 y^{^{''}} + 112 y^{^{'}} + 49 y = 0

\frac{1}{16} x^{^{''}} + 2 x^{^{'}} + x = 0

What is the solution for y^{''}+a^2*n^2*pi^2*y=0 ?
The solution for y^{''}+a^2*n^2*pi^2*y=0 is y=c_{1}cos(piant)+c_{2}sin(piant)