Internal problem ID [10218]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
28. Summary. Page 59
Problem number: Ex 8.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) \left (y^{\prime }\right )^{2}-2 y x y^{\prime }+y^{2}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 47
dsolve((1+x^2)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+y(x)^2-1=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = x c_{1}-\sqrt {-c_{1}^{2}+1} \\ y \relax (x ) = x c_{1}+\sqrt {-c_{1}^{2}+1} \\ y \relax (x ) = c_{1} \sqrt {x^{2}+1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.122 (sec). Leaf size: 73
DSolve[(1+x^2)*(y'[x])^2-2*x*y[x]*y'[x]+y[x]^2-1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x-\sqrt {1-c_1{}^2} \\ y(x)\to c_1 x+\sqrt {1-c_1{}^2} \\ y(x)\to -\sqrt {x^2+1} \\ y(x)\to \sqrt {x^2+1} \\ \end{align*}