What is the solution for y^{''}-(n^2)/4 y=0 ?

The solution for y^{''}-(n^2)/4 y=0 is y=c_{1}e^{(nt)/2}+c_{2}e^{-(nt)/2}

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y^{''}-(n^2)/4 y=0

y^{^{''}} - \frac{n ^{2}}{4} y = 0

y = c_{1} e^{\frac{n t}{2}} + c_{2} e^{- \frac{n t}{2}}

Solution steps

y^{''} (t) - \frac{n ^{2}}{4} y = 0

Solve linear ODE:

y = c_{1} e^{\frac{n t}{2}} + c_{2} e^{- \frac{n t}{2}}

y = c_{1} e^{\frac{n t}{2}} + c_{2} e^{- \frac{n t}{2}}

\frac{d ^{2} y}{d x ^{2}} + 4 (\frac{d y}{d x}) + 4 y = 0

4 x^{^{''}} + 3 x^{^{'}} + 5 x = 0

y^{^{''}} + 64 y = 0, y^{^{'}} (0) = 0

x^{^{''}} + 2^{2} \cdot 3^{2} x = 0

y^{^{''}} + 4 y = 0, y (0) = \frac{1}{2}, y^{^{'}} (0) = 3

What is the solution for y^{''}-(n^2)/4 y=0 ?
The solution for y^{''}-(n^2)/4 y=0 is y=c_{1}e^{(nt)/2}+c_{2}e^{-(nt)/2}