- #1
pk415
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Homework Statement
Ok, so I can get through most of this but I can't seem to get the last part... Here is the problem
[tex]xU_x + (y^2+1)U_y = U-1; U(x,x) = e^x[/tex]
Homework Equations
The Attempt at a Solution
Characteristic equations are:
[tex]\frac{dx}{x} = \frac{dy}{y^2+1} = \frac{dU}{U-1}[/tex]
Solving the first and third gives:
[tex]\frac{U-1}{x} = c_1[/tex]
The first and second equation yield:
[tex]tan^{-1}(y) - lnx = c_2[/tex]
Put the two together in the form
[tex]c_1 = f(c_2)[/tex]
[tex]\frac{U-1}{x} = f(tan^{-1}(y) - lnx)[/tex]
Sub in the Cauchy data and you get
[tex]\frac{e^x-1}{x} = f(tan^{-1}(x) - lnx)[/tex]
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.