What are the Uses of Cauchy Euler equation?

Answer 1

One thing about math is that sometimes the challenge of solving a difficult problem is more rewarding than even it's application to the "real" world. And the applications lead to other applications and new problems come up with other interesting solutions and on and on...

But...

The Cauchy-Euler equation comes up a lot when you try to solve differential equations (the Cauchy-Euler equation is an ordinary differential equation, but more complex partial differential equations can be decomposed to ordinary differential equations); differential equations are used extensively by engineers and scientists to describe, predict, and manipulate real-world scenarios and problems. Specifically, the Cauchy-Euler equation comes up when the solution to the problem is of the form of a power - that is the variable raised to a real power. Specific cases involving equilibrium phenomena - like heat energy through a bar or electromagnetics often rely on partial differential equations (Laplace's Equation, or the Helmholtz equation, for example), and there are cases of these which can be separated into the Cauchy-Euler equation.