69.1.48 problem 67

Internal problem ID [14201]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 67
Date solved : Tuesday, January 28, 2025 at 06:21:40 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

Time used: 0.395 (sec). Leaf size: 53

DSolve[(1-x^2)*D[y[x],x]-x*y[x]+a*x*y[x]^2==0,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 (x^2-1\right )+c_1\right ] \\ y(x)\to 0 \\ y(x)\to \frac {1}{a} \\ \end{align*}