First order pde cauchy problem by method of characteristics

[pk415]

Ok, so I can get through most of this but I can't seem to get the last part... Here is the problem

xU_x + (y^2+1)U_y = U-1; U(x,x) = e^x


Characteristic equations are:

\frac{dx}{x} = \frac{dy}{y^2+1} = \frac{dU}{U-1}

Solving the first and third gives:

\frac{U-1}{x} = c_1

The first and second equation yield:

tan^{-1}(y) - lnx = c_2

Put the two together in the form

c_1 = f(c_2)

\frac{U-1}{x} = f(tan^{-1}(y) - lnx)

Sub in the Cauchy data and you get

\frac{e^x-1}{x} = f(tan^{-1}(x) - lnx)

Now how do I find what my arbitrary function f is? I have spent hours on this. Is there something that relates inverse tan to natural log? Arrggghhhh!

Thanks for any help.

 

[Big-T]

You may have a look here:
http://en.wikipedia.org/wiki/Inverse_trigonometric_function#Logarithmic_forms"