ODE
\[ x \left (a^2 x+y(x) \left (x^2-y(x)^2\right )\right ) y'(x)^2+\left (2 a^2 x y(x)+\left (x^2-y(x)^2\right )^2\right ) y'(x)+a^2 y(x)^2-x \left (x^2-y(x)^2\right ) y(x)=0 \] ODE Classification
[`y=_G(x,y')`]
Book solution method
Change of variable
Mathematica ✗
cpu = 34.7334 (sec), leaf count = 0 , could not solve
DSolve[a^2*y[x]^2 - x*y[x]*(x^2 - y[x]^2) + (2*a^2*x*y[x] + (x^2 - y[x]^2)^2)*Derivative[1][y][x] + x*(a^2*x + y[x]*(x^2 - y[x]^2))*Derivative[1][y][x]^2 == 0, y[x], x]
Maple ✓
cpu = 24.846 (sec), leaf count = 28
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{4}+2\,{x}^{2} \left ( y \left ( x \right ) \right ) ^{2}+8\,{a}^{2}xy \left ( x \right ) +{x}^{4}=0 \right \} \] Mathematica raw input
DSolve[a^2*y[x]^2 - x*y[x]*(x^2 - y[x]^2) + (2*a^2*x*y[x] + (x^2 - y[x]^2)^2)*y'[x] + x*(a^2*x + y[x]*(x^2 - y[x]^2))*y'[x]^2 == 0,y[x],x]
Mathematica raw output
DSolve[a^2*y[x]^2 - x*y[x]*(x^2 - y[x]^2) + (2*a^2*x*y[x] + (x^2 - y[x]^2)^2)*Derivative[1][y][x] + x*(a^2*x + y[x]*(x^2 - y[x]^2))*Derivative[1][y][x]^2 == 0, y[x], x]
Maple raw input
dsolve(x*(a^2*x+(x^2-y(x)^2)*y(x))*diff(y(x),x)^2+(2*a^2*x*y(x)+(x^2-y(x)^2)^2)*diff(y(x),x)+a^2*y(x)^2-x*y(x)*(x^2-y(x)^2) = 0, y(x),'implicit')
Maple raw output
y(x)^4+2*x^2*y(x)^2+8*a^2*x*y(x)+x^4 = 0