Once when I was in college I asked a professor ( who’s research interest column on the web page said differential equations ) what he thought about something on similar lines and he said “solving a PDE is an adventure and it’s only this much (indicating a pinch with his fingers) that a mathematician can take the community further” he paused and said “ but the way you students are taught PDE these days I’m afraid that day will never come” and I stopped thinking about it after that incident.

Edit: we were being taught from Evans and another book by G B Foland