What is the solution for (d^2y)/(dx^2)=16y ?

The solution for (d^2y)/(dx^2)=16y is y=c_{1}e^{4x}+c_{2}e^{-4x}

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(d^2y)/(dx^2)=16y

\frac{d ^{2} y}{d x ^{2}} = 16 y

y = c_{1} e^{4 x} + c_{2} e^{- 4 x}

Solution steps

\frac{d ^{2} y}{d x ^{2}} = 16 y

Solve linear ODE:

y = c_{1} e^{4 x} + c_{2} e^{- 4 x}

y = c_{1} e^{4 x} + c_{2} e^{- 4 x}

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

\frac{d ^{2} y}{d x ^{2}} + 8 \frac{d y}{d x} + 16 y = 0

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

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

y^{^{''}} - 3 y^{^{'}} - 4 y = 0, y (0) = - 1, y^{^{'}} (0) = - 3

What is the solution for (d^2y)/(dx^2)=16y ?
The solution for (d^2y)/(dx^2)=16y is y=c_{1}e^{4x}+c_{2}e^{-4x}