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 = 6.31425 (sec), leaf count = 305
\[\left \{\left \{y(x)\to c_1\right \},\text {Solve}\left [\frac {\left (2-\frac {2 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}\right ) \left (\frac {\frac {6 x^3}{x^2-a y(x)}-4 x}{\sqrt [3]{x^3}}+4\right ) \left (\left (1-\frac {x \left (2 a y(x)+x^2\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}\right ) \log \left (\frac {2-\frac {2 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}}{\sqrt [3]{2}}\right )+\left (\frac {2 a x y(x)+x^3}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}-1\right ) \log \left (\frac {\frac {\frac {6 x^3}{x^2-a y(x)}-4 x}{\sqrt [3]{x^3}}+4}{\sqrt [3]{2}}\right )-3\right )}{18 \sqrt [3]{2} \left (-\frac {\left (2 a y(x)+x^2\right )^3}{\left (x^2-a y(x)\right )^3}+\frac {3 \left (2 a x y(x)+x^3\right )}{\sqrt [3]{x^3} \left (x^2-a y(x)\right )}-2\right )}=c_1+\frac {2\ 2^{2/3} x \log (x)}{9 \sqrt [3]{x^3}},y(x)\right ]\right \}\]
Maple ✓
cpu = 0.02 (sec), leaf count = 37
\[ \left \{ \ln \left ( x \right ) -{\it \_C1}+{\frac {1}{2\,ay \left ( x \right ) } \left ( \ln \left ( {\frac {y \left ( x \right ) }{{x}^{2}}} \right ) ay \left ( x \right ) +{x}^{2} \right ) }=0,y \left ( x \right ) ={\it \_C1} \right \} \] 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),'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