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Consider the Tricomi equation $$yu_{xx} + u_{yy} = 0$$ Find ordinary differential equations describing the real characteristic curves and solve these ODEs to obtain equations for the characteristic curves. Sketch the characteristic curves where they exist.

So Tricomi equation is hyperbolic if $y<0$, parabolic if $y=0$, elliptic if $y>0$. We need to find the real characteristics, so we need to look at the hyperbolic case $y<0$. From the Wikipedia article, we can deduce that the characteristics for this equation are the curves $x \pm \frac{2}{3}(-y)^{3/2}=C$. How to derive the ODEs for characteristics?

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This linear second-order equation rewrites as $$ L[u] = a u_{xx} + 2bu_{xy} + c u_{yy} = 0 $$ where $a = y$, $b = 0$ and $c = 1$. Computing the discriminant $\Delta = b^2 - ac$ tells that the equation is hyperbolic if $\Delta = -y > 0$. We introduce the change of coordinates $(x,y) \mapsto (\xi(x,y),\eta(x,y))$. To obtain the coordinates $\xi$, $\eta$ which reduce the PDE to its canonical form $w_{\xi\eta} = \ell[w]$, we solve the polynomial equation $a \lambda^2 + 2b \lambda + c = 0$, which roots are $\lambda = \pm (-y)^{-1/2}$. The characteristic equations are $$ \frac{\text d x}{\text d t} = 1, \qquad \frac{\text d y}{\text d t} = -\lambda, \qquad \frac{\text d z}{\text d t} = 0 . $$ Therefore, $\frac{\text d y}{\text d x} = \frac{\text d y/\text d t}{\text d x/\text d t} = \mp(-y)^{-1/2}$ gives the expected expression of characteristic curves, along which $\xi$ or $\eta$ is constant.


Further reading: p. 162-163 of

R. Courant, D. Hilbert: Methods of Mathematical Physics vol. II: "Partial differential equations". Wiley-VCH, 1962. doi:10.1002/9783527617234

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