ln(u^2)+u^2=x^2+C?

Question

ln(u^2)+u^2=x^2+C

I am looking for the particular solution, that's u=...... for the above equation and get stuck! could any better brains pls help? thanks.

Answer 1

Let me guess, you got this while solving a separable differential equation, right? Well, unfortunately in some cases it is not possible to find an explicit solution to the differential equation, and you have to leave the unsolved implicit solution. If you're restricted to elementary functions, this is one of those times. If you're allowed to use the Lambert W function however, the simplification becomes easy. First, exponentiate both sides:

u²e^(u²) = e^(x²+C)

Apply the W function:

u² = W(e^(x²+C))

And take the square root:

u=√W(e^(x²+C)) or u=-√W(e^(x²+C))