[l__l]

This makes the same mistake being called out in the comment you're replying to. The point isn't about the mechanics of solving a differential equation, it's about gathering the intuition about a way of approaching problems.

(Also, while it might not be the tools needed for the average homeowner, there are plenty of optimisation problems similar to "how much fence do I need" which are most easily solved by solving the Euler-Lagrange equations)

[DrScientist]

Interesting I've seen the opposite in highly trained mathematicians - they can really struggle with taking an intuitive leap without the safety net of the working out the numbers.

While it's always better to work out the numbers if possible, sometimes it isn't possible and people get trapped trying.