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

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

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

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

y = c_{1} cos (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)

y = c_{1} cos (πan t)

\frac{d ^{2} y}{d x ^{2}} + k^{2} \cdot y^{^{'}} = 0

4 y^{^{''}} + 4 y^{^{'}} + 6 y = 0

\frac{d ^{2} y}{d x ^{2}} - 12 \frac{d y}{d x} + 35 y = 0

y^{^{''}} + y^{^{'}} + 3.25 y = 0

y^{^{''}} - (2 a - 1) y^{^{'}} + a (a - 1) y = 0

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