ODE No. 266

$$\sqrt{x^2+1}(y(x)-x)y^{\prime }(x)-a\sqrt{{(y(x{)}^2+1)}^3}=0$$ ✓ Mathematica : cpu = 1.89272 (sec), leaf count = 69

DSolve[-(a*Sqrt[(1 + y[x]^2)^3]) + Sqrt[1 + x^2]*(-x + y[x])*Derivative[1][y][x] == 0,y[x],x]
 

$$\text{Solve}[\{\frac{2a{\tan }^{-1}\left(\frac{1-a\tan \left(\frac{K[1]}{2}\right)}{\sqrt{a^2-1}}\right)}{\sqrt{a^2-1}}+K[1]+{\tan }^{-1}(x)=c_1,y(x)=\frac{\tan (K[1])+x}{1-x\tan (K[1])}\},\{K[1],y(x)\}]$$ ✓ Maple : cpu = 2.161 (sec), leaf count = 65

dsolve((y(x)-x)*(x^2+1)^(1/2)*diff(y(x),x)-a*((y(x)^2+1)^3)^(1/2) = 0,y(x))
 

$$y\left(x\right)=\tan \left(\mathrm{RootOf}(-\arctan \left(x\right)+{\int }_{}^{-\arctan \left(x\right)+\mathit{\text{\_Z}}}-\frac{-\sqrt{2}\sqrt{\frac{a^2}{\cos \left(2\mathit{\text{\_a}}\right)+1}}\sin \left(2\mathit{\text{\_a}}\right)+\cos \left(2\mathit{\text{\_a}}\right)-1}{2a^2+\cos \left(2\mathit{\text{\_a}}\right)-1}d\mathit{\text{\_a}}+c_1)\right)$$