77.1.63 problem 82 (page 120)

Internal problem ID [17953]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 82 (page 120)
Date solved : Tuesday, January 28, 2025 at 11:18:13 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Solution by Maple

Time used: 0.224 (sec). Leaf size: 57

dsolve(diff(y(x),x)^2*(x^2-1)-2*diff(y(x),x)*x*y(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: 0.110 (sec). Leaf size: 73

DSolve[D[y[x],x]^2*(x^2-1)-2*D[y[x],x]*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 {1-x^2} \\ y(x)\to \sqrt {1-x^2} \\ \end{align*}