4.948   2axy ′(x)− ay(x)+ y(x)y′(x)2 = 0

ODE

2axy′(x)− ay(x)+ y(x )y′(x)2 = 0

ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Homogeneous ODE, xnf (y,y′) = 0
    x

Mathematica
cpu = 0.0867309 (sec), leaf count = 109

{{         ∘ -------------}  {       ∘ --------------} {         ∘ -------------}  {       ∘ --------------}}
   y(x) → −   ec1 (ec1 − 2√ax ) , y(x) → ec1 (ec1 − 2√ax-) , y(x ) → −  ec1 (2√ax-+ ec1) , y(x) →   ec1 (2√ax + ec1)

Maple
cpu = 0.08 (sec), leaf count = 107

{                                                                                                                               }
   2        2           ∫ y(xx)     1    (    2  ∘ --------   )                   ∫ y(xx)     1     (  2  ∘ --------   )
 ax  + (y (x)) = 0,ln(x)−      -a(-a2 +-a) −-a +   a2a + a2 − a d-a−-C1 = 0,ln(x)+       -a( a2-+a) -a +   a2a + a2 + a d-a−-C1 = 0

Mathematica raw input

DSolve[-(a*y[x]) + 2*a*x*y'[x] + y[x]*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -Sqrt[E^C[1]*(E^C[1] - 2*Sqrt[a]*x)]}, {y[x] -> Sqrt[E^C[1]*(E^C[1] - 
2*Sqrt[a]*x)]}, {y[x] -> -Sqrt[E^C[1]*(E^C[1] + 2*Sqrt[a]*x)]}, {y[x] -> Sqrt[E^
C[1]*(E^C[1] + 2*Sqrt[a]*x)]}}

Maple raw input

dsolve(y(x)*diff(y(x),x)^2+2*a*x*diff(y(x),x)-a*y(x) = 0, y(x),'implicit')

Maple raw output

a*x^2+y(x)^2 = 0, ln(x)-Intat((-_a^2+(_a^2*a+a^2)^(1/2)-a)/_a/(_a^2+a),_a = y(x)
/x)-_C1 = 0, ln(x)+Intat((_a^2+(_a^2*a+a^2)^(1/2)+a)/_a/(_a^2+a),_a = y(x)/x)-_C
1 = 0