What is the solution for (d^2x)/(dt^2)+k(dx)/(dt)+a=0 ?

The solution for (d^2x)/(dt^2)+k(dx)/(dt)+a=0 is x=c_{1}+c_{2}e^{-kt}-(at)/k

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(d^2x)/(dt^2)+k(dx)/(dt)+a=0

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

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

Solution steps

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

Solve linear ODE:

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

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

3 \frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 30 y = 15 sin (x)

y^{^{''}} - y^{^{'}} = 3 x

y^{^{''}} - y^{^{'}} + 9 y = 3 sin (3 x)

y^{^{''}} + y^{^{'}} = 4

y^{^{''}} + y^{^{'}} = e^{- x}

What is the solution for (d^2x)/(dt^2)+k(dx)/(dt)+a=0 ?
The solution for (d^2x)/(dt^2)+k(dx)/(dt)+a=0 is x=c_{1}+c_{2}e^{-kt}-(at)/k