What is the solution for (d^2x)/(dt^2)-ax=0 ?

The solution for (d^2x)/(dt^2)-ax=0 is x=c_{1}e^{sqrt(a)t}+c_{2}e^{-sqrt(a)t}

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(d^2x)/(dt^2)-ax=0

\frac{d ^{2} x}{d t ^{2}} - a x = 0

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

Solution steps

\frac{d ^{2} x}{d t ^{2}} - a x (t) = 0

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}

y^{^{''}} - 3 y^{^{'}} - 4 y = 0, y (0) = a, y^{^{'}} (0) = - 10

y^{^{'}} - 2 y^{^{'''}} + y^{^{''}} = 0

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

\frac{d ^{2} x}{d t ^{2}} + 0.75 \frac{d x}{d t} + 0.3 x = 0

y^{^{''}} + 10 y^{^{'}} + 2 y = 0

What is the solution for (d^2x)/(dt^2)-ax=0 ?
The solution for (d^2x)/(dt^2)-ax=0 is x=c_{1}e^{sqrt(a)t}+c_{2}e^{-sqrt(a)t}