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To Zapper. Some differential equations have no analytical solutions but some do. I remember a case where a co worker was using a computer to find the solution to x dot = 1 / (1 + x squared). I asked him, when you finish, are you going to compare your answer with x = inv tan (t). He told me you cannot integrate this equation because the problem is non-linear. Can you beat that?

I was also told that integral exp -ax^2 from 0 to infinity could not be evaluated analytically (his teachers told him). When I showed him the evaluation, he told me make sure to write it down, in case I would forget it. Both these co-workers had advanced degrees in engineering.

These are egregious cases but I can cite at least a few more which are only slightly less egregious.

The computer has opened new vistas in the solutions to differential equations, and I agree only a small minority of problems can be solved in any other way. Nevertheless, I think solving (using separation of variables) e.g. The Hydrogen Atom, The QM harmonic oscillator, Legendre's differential equation, etc, (usually by power series), and understanding why the separat1ion constants are chosen (to terminate the series), should be bread and butter for newly minted physicists.

Knowing the solution to these idealized problems is often a first step in modeling actual problems efficiently.

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# Physicists' knowledge of differential equations

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