Internal problem ID [7846]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 266.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [x=_G(y,y')]
Solve \begin {gather*} \boxed {\left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime }-a \sqrt {\left (y^{2}+1\right )^{3}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 192
dsolve((y(x)-x)*sqrt(x^2+1)*diff(y(x),x)-a*sqrt((y(x)^2+1)^3)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i \\ y \relax (x ) = i \\ y \relax (x ) = \frac {-x +\sqrt {-a^{2} \left (a^{2} x^{4}+2 a^{2} x^{2}-x^{4}+a^{2}-2 x^{2}-1\right )}}{a^{2} x^{2}+a^{2}-1} \\ y \relax (x ) = -\frac {x +\sqrt {-a^{2} \left (a^{2} x^{4}+2 a^{2} x^{2}-x^{4}+a^{2}-2 x^{2}-1\right )}}{a^{2} x^{2}+a^{2}-1} \\ y \relax (x ) = \tan \left (\RootOf \left (-\arctan \relax (x )+\int _{}^{-\arctan \relax (x )+\textit {\_Z}}\frac {\sqrt {2}\, \sqrt {\frac {a^{2}}{\cos \left (2 \textit {\_a} \right )+1}}\, \sin \left (2 \textit {\_a} \right )-\cos \left (2 \textit {\_a} \right )+1}{2 a^{2}+\cos \left (2 \textit {\_a} \right )-1}d \textit {\_a} +c_{1}\right )\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.335 (sec). Leaf size: 69
DSolve[(y[x]-x)*Sqrt[x^2+1]*y'[x]-a*Sqrt[(y[x]^2+1)^3]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{\frac {2 a \text {ArcTan}\left (\frac {1-a \tan \left (\frac {K[1]}{2}\right )}{\sqrt {a^2-1}}\right )}{\sqrt {a^2-1}}+K[1]+\text {ArcTan}(x)=c_1,y(x)=\frac {\tan (K[1])+x}{1-x \tan (K[1])}\right \},\{K[1],y(x)\}\right ] \]