47.1.19 problem 19

Internal problem ID [7400]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number : 19
Date solved : Monday, January 27, 2025 at 02:52:45 PM
CAS classification : [_separable]

\begin{align*} 2 x \sqrt {1-y^{2}}+y^{\prime } y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 27

dsolve(2*x*sqrt(1-y(x)^2)+y(x)*diff(y(x),x)=0,y(x), singsol=all)
 

\[ c_{1} +x^{2}+\frac {\left (y-1\right ) \left (y+1\right )}{\sqrt {-y^{2}+1}} = 0 \]

✓ Solution by Mathematica

Time used: 0.305 (sec). Leaf size: 69

DSolve[2*x*Sqrt[1-y[x]^2]+y[x]*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {-x^4+2 c_1 x^2+1-c_1{}^2} \\ y(x)\to \sqrt {-x^4+2 c_1 x^2+1-c_1{}^2} \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}