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Find the general solution $u(x,y)$ to $$x u_x + y u_y = 0$$

Attempted solution - The characteristic equation satisfy the ODE $dy/dx = y/x$. To solve the ODE, we separate variables: $dy/y = dx/x$; hence $\ln(y) = \ln(x)- C$, so that $$y = x \exp(-C)$$

I am a bit confused in finding the general solution for $u(x,y)$. I just want to know if I am on the right track or not. I haven't taken a formal ODE course but I have self-studied but not in a while. Any suggestions would be greatly appreciated.

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  • $\begingroup$ @Winther Not sure I thought that it was just a constant used. I was following an example from the PDE book by Strauss $\endgroup$ – Wolfy May 18 '17 at 23:13
  • $\begingroup$ I think you mean $u(x,y)$ not $u(x,t)$ $\endgroup$ – Rumplestillskin May 18 '17 at 23:13
  • $\begingroup$ @Rumplestillskin Yes, I believe so too must be an error I will change it. $\endgroup$ – Wolfy May 18 '17 at 23:15
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    $\begingroup$ You need to reread what the method of characteristics tell you to do. The characteristic equation is not correct. It should be $dy/dx = y/x$ which follows from $dy/($coefficient of $u_y$) = $dx/($ coefficient of $u_x$). $\endgroup$ – Winther May 18 '17 at 23:15
  • $\begingroup$ @Winther Could you tell me why $dy.dx = y/x$ $\endgroup$ – Wolfy May 18 '17 at 23:17
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We consider first order PDE's as a first order directional derivative. That is suppose we parametrize a curve $(x,y)$ by a parameter $\xi$. So that

$$ u=u(x(\xi),y(\xi)) $$

$$ \frac{\mathrm{d}u}{\mathrm{d}\xi}=\frac{\mathrm{d}x}{\mathrm{d}\xi}\frac{\partial u}{\partial x}+\frac{\mathrm{d}y}{\mathrm{d}\xi}\frac{\partial u}{\partial y} $$

Comparing to your original PDE

$$ 0 = x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y} $$

We then set

$$ \frac{\mathrm{d}u}{\mathrm{d}\xi}=0 $$

$$ \frac{\mathrm{d}x}{\mathrm{d}\xi}=x $$

$$ \frac{\mathrm{d}y}{\mathrm{d}\xi}=y $$

Along the characteristics $u$ is constant and

$$ \mathrm{d}\xi=\frac{\mathrm{d}x}{x}=\frac{\mathrm{d}y}{y} $$

Integrating we find that $y/x$ is also a constant. Since $u$ is a constant and $y/x$ is a constant, we are free to set one constant equal to an arbitrary function of the other constant.

$$ u(x,y)=\varphi(\frac{y}{x}) $$

where $\varphi$ is an arbitrary differentiable function.

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  • $\begingroup$ I haven't taken much ODE so I'm a bit lost on why $d x/ d\Epsilon = x$? Could you clarify that? $\endgroup$ – Wolfy May 19 '17 at 1:35
  • $\begingroup$ You compare the original PDE to the directional derivative, equate coefficients. $\endgroup$ – Tucker May 19 '17 at 2:02
  • $\begingroup$ Do you know how I would proceed with this if we were asked to find the solution that satisfies the initial condition $u(x,y) = y$ whenever $x^2 + y^2 = 1$? $\endgroup$ – Wolfy May 19 '17 at 5:55
  • $\begingroup$ The boundary condition is addressed below. $\endgroup$ – Tucker May 19 '17 at 6:07
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If

$$ u(x,y)=y $$

when $x^2+y^2=1$

Since $\varphi$ was arbitrary let me define $F$ so that $\varphi(z)=F(z^2)$, our general solution is then

$$ u(x,y)=F(\frac{y^2}{x^2}) $$

Let me know apply the boundary condition

$$ y=F(\frac{y^2}{1-y^2}) $$

Define $\zeta$ so that

$$ \zeta=\frac{y^2}{1-y^2} $$

$$ \zeta=y^2(1+\zeta) $$

$$ y=\sqrt{\frac{\zeta}{1+\zeta}} $$

so that

$$ F(\zeta)=\sqrt{\frac{\zeta}{1+\zeta}} $$

$$ u(x,y)=F(\frac{y^2}{x^2})=\sqrt{\frac{\frac{y^2}{x^2}}{1+\frac{y^2}{x^2}}} $$

$$ u(x,y)=\frac{y}{\sqrt{x^2+y^2}} $$

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