Euler–Tricomi equation

In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow. It is named after mathematicians Leonhard Euler and Francesco Giacomo Tricomi.

${\displaystyle u_{xx}+xu_{yy}=0.\,}$

It is elliptic in the half plane x > 0, parabolic at x = 0 and hyperbolic in the half plane x < 0. Its characteristics are

${\displaystyle x\,dx^{2}+dy^{2}=0,\,}$

which have the integral

${\displaystyle y\pm {\frac {2}{3}}x^{3/2}=C,}$

where C is a constant of integration. The characteristics thus comprise two families of semicubical parabolas, with cusps on the line x = 0, the curves lying on the right hand side of the y-axis.