An underdetermined system of differential equations may have infinitely many solutions--like a single equation with two dependent variables. How can one find any (all?) solutions?
Here's another sense of underdetermined:
DSolve[{x''[t] + 4 x'[t] + 5 x[t] == 0, x[0] + x'[0] == 1}, x[t], t]
Here is a case when Mathematica gives the solutions as a single equation with two dependent variables:
DSolve[y'[t] + Cos[x[t] y[t]] (y[t] x'[t] + x[t] y'[t]) == 0, y[t], t]
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