What is the solution for (d^2x)/(dt^2)=a(x) ?

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

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(d^2x)/(dt^2)=a(x)

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

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

Solution steps

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

Solve linear ODE:

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

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

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

49 y^{^{''}} - 81 y = 0, y (0) = 1, y^{^{'}} (0) = 0

y^{^{''}} + 36 y = 0, y (0) = 4, y^{^{'}} (0) = - 4

y^{^{''''''}} - 64 y = 0

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

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