What is the solution for (d^2y)/(dx^2)=ay ?

The solution for (d^2y)/(dx^2)=ay is y=c_{1}e^{sqrt(a)x}+c_{2}e^{-sqrt(a)x}

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(d^2y)/(dx^2)=ay

\frac{d ^{2} y}{d x ^{2}} = a y

y = c_{1} e^{a x} + c_{2} e^{- a x}

Solution steps

\frac{d ^{2} y}{d x ^{2}} = a y (x)

Solve linear ODE:

y = c_{1} e^{a x} + c_{2} e^{- a x}

y = c_{1} e^{a x} + c_{2} e^{- a x}

y^{^{''}} - 6 y^{^{'}} + 9 y = 0, y^{1} (0) = e^{3} x

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

2 y^{^{'''}} + 9 y^{^{''}} - 6 y^{^{'}} - 5 y = 0

y^{^{''}} + 8 y^{^{'}} + 4 y = 0, y (0) = 0

16 y^{^{''}} + 56 y^{^{'}} + 49 y = 0, y (0) = 5

What is the solution for (d^2y)/(dx^2)=ay ?
The solution for (d^2y)/(dx^2)=ay is y=c_{1}e^{sqrt(a)x}+c_{2}e^{-sqrt(a)x}