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$u_t=\gamma u_{xx}$) admits $ rac{\partial}{\partial x}$, translations along $x$. $( )$ it then follows that the equation has solutions $arphi(x,t)$ and $\psi(x,t)$ such, that $arphi(x,t)=\psi(x-a,t)$, $a=$const i tried reading olver's 'intro to pdes' (see pages , ) but still cannot derive $( ).$ how
− a, x − b). a simple application of the chain rule proves that the partial derivatives of u with respect to t and x agree with the corresponding partial derivatives of u, so $u_t=u_t, u_x=u_x, u_{xx}=u_{xx}$ and so on. in particular, the function u(t, x) is a solution to the heat equation $u_t = \gamma...
https://math.stackexchange.com/questions/3494563/equation-admits-frac-partial-partial-x-and-the-derivation-that-follows