# Characteristic coordinates $ξ(x, y)$ and $η(x, y)$ for $xu_{xx} + u_{yy} = 0$ when $x<0$

How would I determine the characteristic coordinates for $$xu_{xx} + u_{yy} = 0$$?

This PDE reads $$au_{xx} + 2b u_{xy} + cu_{yy} = 0$$ with $$a=x, b=0, c=1$$. The polynomial equation $$a\lambda^2 -2b\lambda +c =0$$ implies $$\lambda^2 = \frac{-1}{x}$$. Since $$x<0$$, we can take $$x=-a$$ where $$a>0$$ and so we get $$\lambda^2 = \frac{1}{a}$$ and thus $$\lambda = dy/dx= \pm \frac{1}{\sqrt{a}}$$ Solving this would give me $$y = \mp 2\sqrt{a} + c = \mp2\sqrt{-x} +c$$, and so $$c = y \pm 2\sqrt{-x}$$. Finally, $$ξ(x, y) = y+2\sqrt{-x} \qquad\text{and}\qquad η(x, y) = y-2\sqrt{-x}$$

Is this correct?

• Further reading: p. 161-162 of R. Courant, D. Hilbert (1962) Methods of Mathematical Physics vol. II: "Partial differential equations", Wiley-VCH. doi:10.1002/9783527617234 – Harry49 Dec 12 '18 at 18:17

Try it out, set $$u(x,y)=v(ξ,η)=v(y+2\sqrt{−x},y-2\sqrt{−x})$$ so that \begin{align} u_x&=-\frac1{\sqrt{-x}}(v_ξ-v_η),& u_y&=v_ξ+v_η\\ u_{xx}&=\frac1{-x}(v_{ξξ}-2v_{ξη}+v_{ηη})+\frac1{2x\sqrt{-x}}(v_ξ-v_η),& u_{yy}&=v_{ξξ}+2v_{ξη}+v_{ηη} \end{align} so that $$xu_{xx}+u_{yy}=4v_{ξη}+\frac1{2\sqrt{-x}}(v_ξ-v_η)$$ which is likely what you wanted to achieve.