Therefore, the complete integral of the equation: F ( u - ln(x), ln(x) - y/x) = 0 or in another form: u = ln(x) + f ( ln(x) - y/x) where f - some differentiable function.
The initial condition u(1,y) = y: u (1, y) = ln(1) + f ( ln(1) - y/1) = f(y) = y Therefore: u (x, y) = ln(x) + ln(x) - y/x = y/x The solution is undefined for x=0. Answer: u (x, y) = y/x, the solution is undefined for x=0.
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