In advanced differential equations we had to solve linear systems of differential equations. The process involved writing it as a matrix, taking a Laplace transform of the matrix, inverting that matrix, taking an inverse Laplace transform of the inverse, and then solving for the initial conditions.
I distinctly remember the experience of trying to solve one problem a dozen times before coming up with the same answer twice. (Unfortunately it wasn't the correct answer...)
That taught me a lot about being careful in algebraic manipulations.
Cheers,
Ben