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

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

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

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

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

Solution steps

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

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

y^{^{''}} + 25 y = 0, y (0) = 4, y^{^{'}} (0) = - 5

y^{^{''}} + 6 y + 9 y = 0

y^{^{'''}} - 6 y^{^{''}} + 12 y^{^{'}} - 8 y = 0

y^{^{''}} + 8 y^{^{'}} + 9 y = 0, y (1) = 1, y^{^{'}} (1) = 0

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