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

The solution for y^{''}-n^2pi^2y=0 is y=c_{1}e^{pint}+c_{2}e^{-pint}

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y^{''}-n^2pi^2y=0

y^{^{''}} - n^{2} π^{2} y = 0

y = c_{1} e^{πn t} + c_{2} e^{- πn t}

Solution steps

y^{''} (t) - n^{2} π^{2} y = 0

Solve linear ODE:

y = c_{1} e^{πn t} + c_{2} e^{- πn t}

y = c_{1} e^{πn t} + c_{2} e^{- πn t}

y^{^{''}} + 2 y^{^{''}} - 3 y = 0

y^{^{''}} - y = 0, y (0) = \frac{1}{5}, y (1) = \frac{1}{5} cosh (1)

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} - 3 y = 0

y^{^{''}} - 2 y^{^{'}} - 8 y = 0, y (0) = 1, y^{^{'}} (0) = - 3

y^{^{'''}} + 2 y^{^{''}} + 5 y^{^{'}} = 0

What is the solution for y^{''}-n^2pi^2y=0 ?
The solution for y^{''}-n^2pi^2y=0 is y=c_{1}e^{pint}+c_{2}e^{-pint}