The Cauchy problem with modified initial data for the generalized Euler–Poisson–Darboux equation

For the equation
$$ \varphi(y-\tau(x))\frac{\partial^2u}{\partial x\partial y}+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}+c(x,y)u=f(x,y), $$
where $\varphi(t)$ is an increasing function with $\varphi(0)=0$, consider the Cauchy problem in different formulations determined by specifying the initial data in various forms on the curve $y=\tau(x)$. It is proved that the problems considered are uniquely solvable.
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