62.12.13 problem Ex 14

Internal problem ID [12859]
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 14
Date solved : Tuesday, January 28, 2025 at 04:28:46 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 22

dsolve((1-x^2)*diff(y(x),x)-x*y(x)=a*x*y(x)^2,y(x), singsol=all)
 

\[ y = \frac {1}{\sqrt {x -1}\, \sqrt {x +1}\, c_{1} -a} \]

✓ Solution by Mathematica

Time used: 0.352 (sec). Leaf size: 57

DSolve[(1-x^2)*D[y[x],x]-x*y[x]==a*x*y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] (a K[1]+1)}dK[1]\&\right ]\left [-\frac {1}{2} \log \left (1-x^2\right )+c_1\right ] \\ y(x)\to 0 \\ y(x)\to -\frac {1}{a} \\ \end{align*}