The Riccati equation $y'(x)+y(x)^2=f(x)$ is non-linear, but can be transformed into the linear equation $-u''(x)+f(x)u(x)=0$ via $y(x)=\frac{u'(x)}{u(x)}$.
Is there a general statement known about when a non-linear first-order ODE $a(y(x),y'(x))+b(y(x))=f(x)$ can be transformed into a linear second-order ODE $-u''(x)+p(x)u'(x)+q(x)u(x)=r(x)$?
For first order equations, of the form $y'=R(x,y)$, $R$ is rational in $y,y'$, analytic in $x$, Riccati is the only case. The general form of a Riccati equation is $$y'=a(x)y^2+b(x)y+c(x).$$ This is due to L. Fuchs. For the proofs, see Ince, Ordinary differential equations, or Mishihiko Matsuda, First-order algebraic differential equations. A differential algebraic approach. Lecture Notes in Mathematics, 804. Springer, Berlin, 1980.
There is also a complete classification for second order equations of the form $y''=R(y',y,x)$ which was obtained by B. Gambier, Acta Math., 33 (1910). There is a list of 50 equations, most of them are reducible to linear or first order equations, and 6 are not (these 6 can be also solved using linear ODE, but by a more complicated correspondence than a simple change of the variable).
See also Kamke, Differentialgleichungen. Losungsmethoden und Losungen, Leipzig, 1959. It reproduces Gambier's list.
