The second order Lagerstrom ODE can be reduced to a first order ODE of Abel type once the system succeeds in finding one polynomial symmetry for it (see ?symgen):

From which, giving the same indication directly to dsolve, you obtain the reduction of order

For the structure of the solution above see ?ODESolStruc. Reductions of order can also be tested with odetest

The reduced ODE is of Abel type and can be selected using the mouse, or as follows

DEtools, odeadvisor, dsolve, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, missing, reducible, linear_ODEs, exact_linear, exact_nonlinear, sym_Fx, linear_sym, Bessel, Painleve, Halm, Gegenbauer, Duffing, ellipsoidal, elliptic, erf, Emden, Jacobi, Hermite, Lagerstrom, Laguerre, Liouville, Lienard, Van_der_Pol, Titchmarsh; for other differential orders see odeadvisor,types.