Implicit solution of ODE to explicit or approximate explicit function

Question

Working with the following ODE and implicit solution but need an explicit solution for J: The ODE with$J_c$ and $G$ as constants is: $$-\frac{1}{J^2}\frac{dJ}{dt} = G(J-J_c)$$

The implicit solution given by Field et al. (1995) is: $$Gt = \frac{1}{J_c^2} \left[ ln \left( \frac{J}{J_o}\cdot \frac{J_o-J_c}{J-J_c} \right) - J_c \left(\frac{1}{J}- \frac{1}{J_o} \right) \right] $$

There is no explicit statement about $J_o$ in the reference but physically it corresponds to the initial flux at time zero. This suggests that $J=J_o \text{ at time}~ t=0$. However as suggested and shown by JJacquelin when we plug in for $t=0$ we get $ln(1) = 0$ ? Checking the -ve sign in the original reference as suggested by JJacquelin.

Any pointers to a complete explicit solution or good approximate of an explicit solution is greatly appreciated. Thanks, Vince

Answer 1

Based on contribution from JJacquelin the problem needs a numerical approach to solve. No solution based on standard functions (possibly non-standard) is likely.

Answer 2

Are you sure that the signs are correct in the given equations ?

As they are presently, the condition $J(t=0)=J_0$ is impossible to fullfil.

With the oposite sign in the first given equation, the solution satisfies the condition.

Bytheway, the mistake of sign could be in my calculus. But I don't think so after checking. Check all by yourself and try to find where is the mistake, in the given equations or in the calculus.