Answer to Question #204414 in Calculus for smi

Question

Q. What is unique solution of a differential equation or a physical problem? Discuss different cases.

Answer 1

There is a relationship between symmetries, conservation laws, and boundary conditions (in differential equations). If you have enough symmetries (i.e., enough conservation laws), you will have a unique solution.

If you had a system with some free parameters and no physics left to describe, you would claim that the physics is taking place in the complete space that your equations function in modulo the free parameters. There would also be a set of equations that exclusively act on that quotient space.

If you don't have a boundary condition to fix the integration constant, then the dimension has survived, and the integration was meaningless.