# Implicit solution of ODE to explicit or approximate explicit function

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

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.

• Checked the reference and the -ve sign is there however it could be a typo in the reference. Need to double check the derivation in the reference. Haven't gone through your solution yet ...but thanks very much for all your work!! Vince – Vince Sep 5 '15 at 21:51
• I went through your solution but not quite sure how to use your answer to isolate for J in terms of the other terms. We still have an implicit expression in J. Maybe it's not possible to separate out (I've tried for some time now) and we need to try an approximate solution. Thanks! – Vince Sep 5 '15 at 22:18
• The inverse of the function $t(J)$ cannot be expressed as a finite number of standard functions. In this case, any further calculus will use numerical methods : either numerical solving of the equation $t(J)=t$ for $J$ given $t$, or direct numerical solving of the ODE, or series expansion which anyways finally requires numerical computation for practical use. – JJacquelin Sep 6 '15 at 7:00
• I've decided to go with numerical solving of the ODE ....Thanks very much for all your help !!! – Vince Sep 6 '15 at 11:36

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