title: Euler–Tricomi equation
text: # 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.
  
      $u_{xx}+xu_{yy}=0.\,$
  
  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
  
  A general expression for particular solutions to the Euler–Tricomi equations is:
  
      $u_{k,p,q}=\sum _{i=0}^{k}(-1)^{i}{\frac {x^{m_{i}}y^{n_{i}}}{c_{i}}}\,$
  
  where
  
      $k\in \mathbb {N}$
      $p,q\in \{0,1\}$
      $m_{i}=3i+p$
      $n_{i}=2(k-i)+q$
      $c_{i}=m_{i}!!!\cdot (m_{i}-1)!!!\cdot n_{i}!!\cdot (n_{i}-1)!!$
  
  These can be linearly combined to form further solutions such as:
  
  for k = 0:
  
      $u=A+Bx+Cy+Dxy\,$
  
  for k = 1:
  
      $u=A({\tfrac {1}{2}}y^{2}-{\tfrac {1}{6}}x^{3})+B({\tfrac {1}{2}}xy^{2}-{\tfrac {1}{12}}x^{4})+C({\tfrac {1}{6}}y^{3}-{\tfrac {1}{6}}x^{3}y)+D({\tfrac {1}{6}}xy^{3}-{\tfrac {1}{12}}x^{4}y)\,$
  
  etc.
  
  The Euler–Tricomi equation is a limiting form of Chaplygin's equation.
  
  