62.17.8 problem Ex 8

Internal problem ID [12908]
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
Date solved : Tuesday, January 28, 2025 at 04:43:01 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

\begin{align*} \left (x^{2}+1\right ) {y^{\prime }}^{2}-2 x y y^{\prime }+y^{2}-1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.158 (sec). Leaf size: 57

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 &= \sqrt {x^{2}+1} \\ y &= -\sqrt {x^{2}+1} \\ y &= c_{1} x -\sqrt {-c_{1}^{2}+1} \\ y &= c_{1} x +\sqrt {-c_{1}^{2}+1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.112 (sec). Leaf size: 73

DSolve[(1+x^2)*(D[y[x],x])^2-2*x*y[x]*D[y[x],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*}