4.20.14 $(x^2-ay(x))y^{\prime }(x{)}^2-2xy(x)y^{\prime }(x)=0$

ODE
$$(x^2-ay(x))y^{\prime }(x{)}^2-2xy(x)y^{\prime }(x)=0$$ ODE Classification

[_quadrature]

Book solution method
Change of variable

Mathematica ✓
cpu = 6.31425 (sec), leaf count = 305

$$\{\{y(x)\to c_1\},\text{Solve}[\frac{(2-\frac{2(2axy(x)+x^3)}{\sqrt[3]{x^3}(x^2-ay(x))})(\frac{\frac{6x^3}{x^2-ay(x)}-4x}{\sqrt[3]{x^3}}+4)((1-\frac{x(2ay(x)+x^2)}{\sqrt[3]{x^3}(x^2-ay(x))})\log \left(\frac{2-\frac{2(2axy(x)+x^3)}{\sqrt[3]{x^3}(x^2-ay(x))}}{\sqrt[3]{2}}\right)+(\frac{2axy(x)+x^3}{\sqrt[3]{x^3}(x^2-ay(x))}-1)\log \left(\frac{\frac{\frac{6x^3}{x^2-ay(x)}-4x}{\sqrt[3]{x^3}}+4}{\sqrt[3]{2}}\right)-3)}{18\sqrt[3]{2}(-\frac{{(2ay(x)+x^2)}^3}{{(x^2-ay(x))}^3}+\frac{3(2axy(x)+x^3)}{\sqrt[3]{x^3}(x^2-ay(x))}-2)}=c_1+\frac{22^{2/3}x\log (x)}{9\sqrt[3]{x^3}},y(x)]\}$$

Maple ✓
cpu = 0.02 (sec), leaf count = 37

$$\{\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}+\frac{1}{2ay\left(x\right)}(\ln \left(\frac{y\left(x\right)}{x^2}\right)ay\left(x\right)+x^2)=0,y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}\}$$ 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),'implicit')

Maple raw output

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