Chain Rule for PDEs

Question

I'm having a hard time seeing how to use the chain rule to prove these two statements and understanding how they work for PDEs in general

Given $D>0$

$u(x,t)$ satisfies $u_{t} = Du_{xx}$ iff $v(y,t) = u(\sqrt{D}y,t)$ satisfies $v_{t} = v_{yy}$

$u(x,t)$ satisfies $u_{t} = Du_{xx}$ iff $v(x,\tau) = u(x,\tau/D)$ satisfies $v_{\tau} = v_{xx}$

Any help would be greatly appreciated

Answer 1

Example Solution

$v_t = \frac{\partial}{\partial t}(u(\sqrt{Dy},t))=\frac{\partial u(\sqrt{Dy},t)}{\partial t}*\frac{\partial}{\partial t}(u)=u_t(\sqrt{Dy},t)$

$v_{yy} = \frac{\partial}{\partial y}(\sqrt{D}\frac{\partial u}{\partial t})=Du_{xx}$