47.2.51 problem 47

Internal problem ID [7467]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 47
Date solved : Monday, January 27, 2025 at 03:01:12 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 38

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

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

✓ Solution by Mathematica

Time used: 0.755 (sec). Leaf size: 62

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

\begin{align*} y(x)\to -\frac {2}{\sqrt {x \left (\log ^2(x)-2 c_1 \log (x)+4+c_1{}^2\right )}} \\ y(x)\to \frac {2}{\sqrt {x \left (\log ^2(x)-2 c_1 \log (x)+4+c_1{}^2\right )}} \\ y(x)\to 0 \\ \end{align*}