dv/dx=a/v

 $$\frac{dv}{dx}=\frac{d}{dt}\log v=\frac{a}{v}$$

We want to find the gravitational potential

$$\frac{d^2\phi}{dx^2}=0$$

in terms of \(v\).

First, note that

$$\frac{dv}{dx}=\frac{d}{dt}\log v=\frac{dv/dt}{v}=\frac{a}{v},$$

$$\frac{da}{dv}=\frac{d}{dt}\log a=\frac{da/dt}{a}=\frac{j}{a}.$$

Then

$$\frac{d\phi}{dx}=\frac{d\phi}{dv}\frac{dv}{dx}=\frac{d\phi}{dv}\frac{a}{v},$$

Apply Laplace equation to get

$$\frac{d}{dv}\left(\frac{d\phi}{dx}\right)=0$$

$$\frac{d}{dv}\left(\frac{d\phi}{dx}\right)=\frac{d}{dv}\left(\frac{d\phi}{dv}\frac{a}{v}\right)=0$$

Assuming \(j=0\) we have the Cauchy-Euler equation

$$v\frac{d^2\phi}{dv^2}-\frac{d\phi}{dv}=0$$

So

$$\phi(v)=Av^2 + B$$