Let the convection equation $\frac{\delta u}{\delta t} + c\frac{\delta u}{\delta x} = 0$. I want to show that $$\frac{\delta^2 u}{\delta t^2}-c\frac{\delta^2 u}{\delta x^2}=0$$
I think it would be possible to differentiate the original convection equation with respect to t and x. Then the following two equations can be obtained. $$\frac{\delta^2 u}{\delta t^2} + c\frac{\delta^2 u}{\delta t \delta x}=0$$ $$\frac{\delta^2 u}{\delta x \delta t} + c\frac{\delta^2 u}{\delta x^2}=0$$ If the two mixed derivatives can be interchanged, I can conclude $\frac{\delta^2 u}{\delta t^2}-c\frac{\delta^2 u}{\delta x^2}=0$ by subtracting above two equations, but I don't know if it is possible to interchange it. Is it possible?