83.22.24 problem 24

Internal problem ID [19278]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 24
Date solved : Tuesday, January 28, 2025 at 01:22:12 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 \end{align*}

✓ Solution by Maple

Time used: 0.060 (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,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.106 (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,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*}