60.1.443 problem 446

Internal problem ID [10457]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 446
Date solved : Monday, January 27, 2025 at 07:46:05 PM
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: 0.080 (sec). Leaf size: 57

dsolve((x^2+1)*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: 0.113 (sec). Leaf size: 73

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