Riccati equation
/ ree-KAH-tee /
A Riccati equation is the first-order equation that is genuinely nonlinear because of a y-squared term, and unlike the Bernoulli or homogeneous cases there is no substitution that solves it from scratch. It sits right at the edge of what elementary methods can do — and it teaches an important honest lesson: sometimes the best you can do is leverage one solution you already happen to know.
Its standard form is y' = q0(x) + q1(x) y + q2(x) y^2. The y^2 term is what makes it nonlinear and resistant. The key fact is conditional: IF you can find (or guess, or are given) one particular solution y1(x), THEN the substitution y = y1 + 1/v reduces the Riccati equation to a first-order LINEAR equation in v, which you solve by an integrating factor. (Equivalently y = y1 + u turns it into a Bernoulli equation in u with n = 2.) So one known solution is the whole game — it converts the unsolvable into the routine.
Why does it matter beyond being a curiosity? Because the Riccati equation is the bridge between first-order nonlinear equations and second-order linear ones: the substitution y = -w'/(q2 w) turns it into a linear second-order equation for w, and conversely every linear second-order equation hides a Riccati equation. It appears in control theory (the matrix Riccati equation is central to optimal control), in quantum mechanics, and in geometry — far more important than its modest appearance suggests.
For y' = y^2 - 2/x^2 one can spot the particular solution y1 = -1/x. Substituting y = -1/x + 1/v and simplifying gives a linear equation v' + (2/x) v = 1, whose solution combines with y1 to give the full one-parameter family. Without that lucky y1, the equation is not solvable by elementary means.
One known solution y1 plus y = y1 + 1/v turns a Riccati equation linear.
There is no general method to find the first particular solution y1 — that step usually relies on a clever guess or special structure. Without a known y1 a Riccati equation typically has no closed-form solution at all.