\[ \left (y(x) y'(x)+x\right ) \left (\frac {x^2}{a}+\frac {y(x)^2}{b}\right )+\frac {(a-b) \left (y(x) y'(x)-x\right )}{a+b}=0 \] ✓ Mathematica : cpu = 0.903245 (sec), leaf count = 204
DSolve[((a - b)*(-x + y[x]*Derivative[1][y][x]))/(a + b) + (x^2/a + y[x]^2/b)*(x + y[x]*Derivative[1][y][x]) == 0,y[x],x]
\[\left \{\left \{y(x)\to -\frac {\sqrt {b} \sqrt {a^2+2 a^2 W\left (\frac {c_1 (a+b) e^{\frac {b x^2}{2 a^2}-\frac {b}{2 a}-\frac {x^2}{2 b}-\frac {1}{2}}}{2 a^3 b^2}\right )+a b-a x^2-b x^2}}{\sqrt {a} \sqrt {a+b}}\right \},\left \{y(x)\to \frac {\sqrt {b} \sqrt {a^2+2 a^2 W\left (\frac {c_1 (a+b) e^{\frac {b x^2}{2 a^2}-\frac {b}{2 a}-\frac {x^2}{2 b}-\frac {1}{2}}}{2 a^3 b^2}\right )+a b-a x^2-b x^2}}{\sqrt {a} \sqrt {a+b}}\right \}\right \}\] ✓ Maple : cpu = 1.201 (sec), leaf count = 240
dsolve((y(x)^2/b+x^2/a)*(y(x)*diff(y(x),x)+x)+(a-b)/(a+b)*(y(x)*diff(y(x),x)-x) = 0,y(x))
\[y \left (x \right ) = \frac {\sqrt {a \left ({\mathrm e}^{\frac {-2 \operatorname {LambertW}\left (\frac {\left (a +b \right ) {\mathrm e}^{-\frac {x^{2}}{2 b}} {\mathrm e}^{\frac {b \,x^{2}}{2 a^{2}}} {\mathrm e}^{-\frac {1}{2}} {\mathrm e}^{-\frac {b}{2 a}} {\mathrm e}^{-\frac {c_{1}}{a b}}}{2 a^{2} b}\right ) a^{2} b +\left (-x^{2}-b \right ) a^{2}+\left (-b^{2}-2 c_{1}\right ) a +b^{2} x^{2}}{2 b \,a^{2}}}+b \left (-x^{2}+a \right )\right )}}{a}\]