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Physical relevancy of two curious PDE's

My research has brought me to the following linear parabolic second order PDE: $$ \frac{\partial^2}{\partial x^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial t}\Psi(t,x) $$ for $c(t,x)=-\frac{t}{x}$ and $...
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