Internal problem ID [8602]
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')`]
\[ \boxed {\left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime }-a \sqrt {\left (1+y^{2}\right )^{3}}=0} \]
✓ Solution by Maple
Time used: 0.234 (sec). Leaf size: 187
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 \left (x \right ) &= \frac {-x +\sqrt {-a^{2} \left (x^{2}+1\right )^{2} \left (a^{2}-1\right )}}{a^{2} x^{2}+a^{2}-1} \\ y \left (x \right ) &= \frac {-x -\sqrt {-a^{2} \left (x^{2}+1\right )^{2} \left (a^{2}-1\right )}}{a^{2} x^{2}+a^{2}-1} \\ \frac {\sqrt {2}\, \sqrt {\frac {a^{2}}{1+\cos \left (2 \arctan \left (x \right )-2 \arctan \left (y \left (x \right )\right )\right )}}\, \cos \left (\arctan \left (x \right )-\arctan \left (y \left (x \right )\right )\right ) \arctan \left (\frac {\cos \left (\arctan \left (x \right )-\arctan \left (y \left (x \right )\right )\right )}{\sqrt {a^{2}-1}}\right )+\arctan \left (\frac {\sqrt {a^{2}-1}\, \tan \left (\arctan \left (x \right )-\arctan \left (y \left (x \right )\right )\right )}{a}\right ) a -\sqrt {a^{2}-1}\, \left (c_{1} -\arctan \left (y \left (x \right )\right )\right )}{\sqrt {a^{2}-1}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.935 (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]
\[ \text {Solve}\left [\left \{\frac {2 a \arctan \left (\frac {1-a \tan \left (\frac {K[1]}{2}\right )}{\sqrt {a^2-1}}\right )}{\sqrt {a^2-1}}+K[1]+\arctan (x)=c_1,y(x)=\frac {\tan (K[1])+x}{1-x \tan (K[1])}\right \},\{K[1],y(x)\}\right ] \]