62.12.1 problem Ex 1

Internal problem ID [12847]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 19. Summary. Page 29
Problem number : Ex 1
Date solved : Tuesday, January 28, 2025 at 04:26:54 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 40

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

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

✓ Solution by Mathematica

Time used: 3.572 (sec). Leaf size: 77

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

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