Could anyone solve this equation ?
Actually, you have a system of four equations in u, v, n, and h. I notice, in particular, that the second and third equations have v both in and exponent and not in an exponent. There is no way to solve such an equation in terms of "elementary functions". I recommend a numerical solution.
Yeah, this one's hopeless when it comes to traditional system methods. Here's a suggestion for a numerical approach:
Equations 2,3,4 are all of the form , for the variable x being n,m,h respectively, which reduces to . So, isolate these in terms of v and sub them in equation 1.
where
You can now sub eqn 1 for n,m,h so you have an equation involving only v equaling zero. This is way too hairy to simplify and isolate v, but you can easily compute values for f(v) and approximate the zeroes of the function. On a graph, it looks like f(v) is an increasing function but at a decreasing rate, crossing the x-axis around x~1. Assuming this is the only solution, you should be able to write a small program to approximate it to any arbitrary level of accuracy. The closest I can get for you with my handheld calculator is x=.8607 plus or minus .00006.