Questions without answers in Partial Differential Equations

Consider the following partial differential equation (PDE)$$(y-1) \frac{\partial^{2} u}{\partial x^{2}}-(x-3)^{2} \frac{\partial^{2} u}{\partial y^{2}}+y^{2} \frac{\parti...

​​​​Consider the second order Partial Differential Equation $\textsf{(PDE)}$$$4 x^{2} \frac{\partial^{2} u}{\partial x^{2}}+4(x+y) \frac{\partial^{2} u}{\partial x \parti...

Let $u(x, t)$ be the solution of the initial boundary value problem$$ \begin{array}{c}\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}-u=0, \quad 0< x ...

Let $u(x, t)$ be the solution of the initial boundary value problem$\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=0, \quad x \in(0,2), t>0$$u(x, 0)=...

If the partial differential equation$$(x+2) \frac{\partial^2 u}{\partial x^2}+2(x+y) \frac{\partial^2 u}{\partial x \partial y}+2(y-1) \frac{\partial^2 u}{\partial y^2}-3...