(NOTE: I am not asking for the solution to this PDE)

I have the PDE, $$u_x u_y + ln(x^2)=0,$$ with the condition that, $$u(x_0,y)=y,$$ where $x_0$ is a constant. I am to find the explicit solution (using Charpit's equations). In order to do this I first need to parameterize the initial condition in the form of a curve $\Gamma (s)$ in order to solve.

I am used to parameterizing curves but not of this type.

I have written down, $$u=s, x=x_0, y=s,$$ just as a hunch but I have no working to show for it. Is this correct? How do you parameterize equations of this type?

Thanks in advance.

  • $\begingroup$ does $\ln(x^2) = 2\ln(x)$ help at all $\endgroup$ – phdmba7of12 Apr 6 at 14:46
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    $\begingroup$ @phdmba7of12 maybe; when solving the equation. Not for the parameterization. Solving the PDE becomes fairly trivial once I have the parameterization. $\endgroup$ – Dan Elsender Apr 6 at 14:54
  • $\begingroup$ what specific text are you referring to describing this method $\endgroup$ – phdmba7of12 Apr 6 at 14:59
  • $\begingroup$ a precedent exists here on the stack math.stackexchange.com/questions/2674756/… $\endgroup$ – phdmba7of12 Apr 6 at 15:00
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    $\begingroup$ @phdmba7of12 I'm asking how to parameterize the initial condtion, not how to solve the PDE. $\endgroup$ – Dan Elsender Apr 6 at 15:05

The correct parameterisation is $$u=s, y=s, x=x_0$$ This leads to a solution through Charpit's method of $$u=-2xln(x)+2x+2x_0ln(x_0)-2x_0+y$$


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