\[ y'(x) \left (\sin (\alpha ) \left (y(x)^2-x^2\right )-2 x \cos (\alpha ) y(x)+\sqrt {x^2+y(x)^2} y(x)\right )+\cos (\alpha ) \left (y(x)^2-x^2\right )+2 x \sin (\alpha ) y(x)+x \sqrt {x^2+y(x)^2}=0 \] ✓ Mathematica : cpu = 5.83406 (sec), leaf count = 116
DSolve[2*x*Sin[alpha]*y[x] + Cos[alpha]*(-x^2 + y[x]^2) + x*Sqrt[x^2 + y[x]^2] + (-2*x*Cos[alpha]*y[x] + Sin[alpha]*(-x^2 + y[x]^2) + y[x]*Sqrt[x^2 + y[x]^2])*Derivative[1][y][x] == 0,y[x],x]
\[\text {Solve}\left [\sqrt {\cos ^2(\alpha )} \sec (\alpha ) \left (\log \left (\cos (\alpha ) \left (\frac {\cos (\alpha ) y(x)}{x}+\sin (\alpha )\right )\right )-\log \left (\frac {1}{2} \left (-2 \sqrt {\cos ^2(\alpha )} \sqrt {\frac {y(x)^2}{x^2}+1}-\frac {\sin (2 \alpha ) y(x)}{x}+\cos (2 \alpha )+1\right )\right )\right )-\frac {1}{2} \log \left (\left (\frac {\cos (\alpha ) y(x)}{x}+\sin (\alpha )\right )^2\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )=-\log (x)+c_1,y(x)\right ]\] ✓ Maple : cpu = 0.725 (sec), leaf count = 129
dsolve((y(x)*(y(x)^2+x^2)^(1/2)+(y(x)^2-x^2)*sin(alpha)-2*x*y(x)*cos(alpha))*diff(y(x),x)+x*(y(x)^2+x^2)^(1/2)+2*x*y(x)*sin(alpha)+(y(x)^2-x^2)*cos(alpha) = 0,y(x))
\[y \left (x \right ) = \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-\textit {\_a}^{3} \cos \left (2 \alpha \right )-3 \textit {\_a}^{2} \sin \left (2 \alpha \right )-\textit {\_a}^{3}+3 \textit {\_a} \cos \left (2 \alpha \right )+\sin \left (2 \alpha \right )+\sqrt {2}\, \sqrt {\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} \cos \left (2 \alpha \right )+2 \textit {\_a} \sin \left (2 \alpha \right )+\textit {\_a}^{2}-\cos \left (2 \alpha \right )+1\right )}-\textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} \cos \left (2 \alpha \right )+2 \textit {\_a} \sin \left (2 \alpha \right )+\textit {\_a}^{2}-\cos \left (2 \alpha \right )+1\right )}d \textit {\_a} +c_{1}\right ) x\]