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Given a differential equation for velocity, $dv/dt + v = 1$, as well as its solution, is it possible to derive a differential equation for velocity with respect to distance?

I found a solution to the differential equation to be

$$v = 1 - 1/e^t$$

and since $dx/dt = v$, $x = t + 1/e^t - 1$.

From this point I have tried to use the chain rule $$\frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt}$$ and ended up getting $dv/dx = (1 - v)/(1 - 1/e^t)$. I am not sure how to get the differential equation in terms of velocity as a function of distance.

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  • $\begingroup$ @DavidZ See my edit. $\endgroup$
    – Jon Martin
    Oct 13, 2015 at 1:06

1 Answer 1

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$$\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}=\frac{dv}{dx}v$$

So the differential equation with x as the independent variable becomes $$v(v'+1)=1$$

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