Clairaut's equation
/ klay-ROH /
Clairaut's equation is a small, strange first-order equation that does something none of the earlier ones do: it has a tidy family of straight-line solutions AND one extra curved solution that is not a member of that family. It is the honest place in the first-order story where the neat picture 'the general solution captures everything' quietly breaks down.
Its form is y = x y' + f(y'), where the unknown's derivative y' appears both linearly (multiplied by x) and inside some function f. The trick to solving it is to differentiate the whole equation with respect to x. After cancelling, you are left with y'' times (x + f'(y')) = 0, a product equal to zero, so one of the two factors must vanish. If y'' = 0, then y' is a constant, say m, and plugging back gives the general solution y = m x + f(m) — a one-parameter family of straight lines, one for each slope m. If instead x + f'(y') = 0, you get a separate curve that does not belong to the line family at all.
That second branch is a singular solution, and geometrically it is the envelope of the family of lines — the curve each line touches tangentially, like the smooth boundary traced out by a moving ruler. Clairaut's equation is therefore the standard textbook window onto the fact that a general solution need not be complete: some solutions can live outside the parameterized family entirely.
Take y = x y' + (y')^2, so f(p) = p^2. The line solutions are y = m x + m^2 for every constant m. The singular branch comes from x + f'(y') = x + 2 y' = 0, i.e. y' = -x/2; substituting back gives the parabola y = -x^2/4, which is tangent to every one of those lines.
A whole family of tangent lines, plus the parabola they all touch — its envelope.
The straight-line family is the general solution, yet it misses the singular solution entirely. So 'general solution' here does NOT mean 'all solutions' — a warning that carries far beyond Clairaut's equation.