##### 4.20.14 $$\left (x^2-a y(x)\right ) y'(x)^2-2 x y(x) y'(x)=0$$

ODE
$\left (x^2-a y(x)\right ) y'(x)^2-2 x y(x) y'(x)=0$ ODE Classiﬁcation

[_quadrature]

Book solution method
Change of variable

Mathematica
cpu = 6.44996 (sec), leaf count = 305

$\left \{\{y(x)\to c_1\},\text {Solve}\left [\frac {\left (2-\frac {2 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}\right ) \left (\frac {\frac {6 x^3}{x^2-a y(x)}-4 x}{\sqrt [3]{x^3}}+4\right ) \left (\left (1-\frac {x \left (2 a y(x)+x^2\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}\right ) \log \left (\frac {2-\frac {2 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}}{\sqrt [3]{2}}\right )+\left (\frac {2 a x y(x)+x^3}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}-1\right ) \log \left (\frac {\frac {\frac {6 x^3}{x^2-a y(x)}-4 x}{\sqrt [3]{x^3}}+4}{\sqrt [3]{2}}\right )-3\right )}{18 \sqrt [3]{2} \left (-\frac {\left (2 a y(x)+x^2\right )^3}{\left (x^2-a y(x)\right )^3}+\frac {3 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}-2\right )}=\frac {2\ 2^{2/3} x \log (x)}{9 \sqrt [3]{x^3}}+c_1,y(x)\right ]\right \}$

Maple
cpu = 0.774 (sec), leaf count = 28

$\left [y \left (x \right ) = -\frac {x^{2}}{a \LambertW \left (-\frac {\textit {\_C1} \,x^{2}}{a}\right )}, y \left (x \right ) = \textit {\_C1}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> C[1]}, Solve[((2 - (2*(x^3 + 2*a*x*y[x]))/((x^3)^(1/3)*(x^2 - a*y[x]))
)*(4 + (-4*x + (6*x^3)/(x^2 - a*y[x]))/(x^3)^(1/3))*(-3 + Log[(2 - (2*(x^3 + 2*a
*x*y[x]))/((x^3)^(1/3)*(x^2 - a*y[x])))/2^(1/3)]*(1 - (x*(x^2 + 2*a*y[x]))/((x^3
)^(1/3)*(x^2 - a*y[x]))) + Log[(4 + (-4*x + (6*x^3)/(x^2 - a*y[x]))/(x^3)^(1/3))
/2^(1/3)]*(-1 + (x^3 + 2*a*x*y[x])/((x^3)^(1/3)*(x^2 - a*y[x])))))/(18*2^(1/3)*(
-2 - (x^2 + 2*a*y[x])^3/(x^2 - a*y[x])^3 + (3*(x^3 + 2*a*x*y[x]))/((x^3)^(1/3)*(
x^2 - a*y[x])))) == C[1] + (2*2^(2/3)*x*Log[x])/(9*(x^3)^(1/3)), y[x]]}

Maple raw input

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

Maple raw output

[y(x) = -1/a/LambertW(-1/a*_C1*x^2)*x^2, y(x) = _C1]