# 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 for Leonhard Euler and Francesco Giacomo Tricomi.

$u_{xx}+xu_{yy}=0.\,$ It is elliptic in the half plane x > 0, parabolic at x = 0 and hyperbolic in the half plane x < 0. Its characteristics are

$x\,dx^{2}+dy^{2}=0,\,$ which have the integral

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

## Particular solutions

Particular solutions to the Euler–Tricomi equations include

• $u=Axy+Bx+Cy+D,\,$
• $u=A(3y^{2}+x^{3})+B(y^{3}+x^{3}y)+C(6xy^{2}+x^{4})+D(2xy^{3}+x^{4}y),\,$

where ABCD are arbitrary constants.

A general expression for these solutions is:

• $u=\sum _{i=0}^{k}{\frac {x^{m_{i}}\cdot y^{n_{i}}}{c_{i}}}\,$

where

• $p,q\in [0,1]$
• $m_{i}=3i+p$
• $n_{i}=2(k-i)+q$
• $c_{i}=m_{i}!!!\cdot (m_{i}-1)!!!\cdot n_{i}!!\cdot (n_{i}-1)!!$

The Euler–Tricomi equation is a limiting form of Chaplygin's equation.