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# 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.

uxx = xuyy.

It is hyperbolic in the half plane x > 0 and elliptic in the half plane x < 0. Its characteristics are xdx2 = dy2, which have the integral $y\pm\frac{2}{3}x^{\frac{3}{2}}=C$

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(3y2 + x3) + B(y3 + x3y) + C(6xy2 + x4)

where A,B,C,D are arbitrary constants.

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

## Bibliography

• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.