32.4.19 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.11, page 90

Internal problem ID [5830]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 10
Problem number : Recognizable Exact Differential equations. Integrating factors. Exercise 10.11, page 90
Date solved : Monday, January 27, 2025 at 01:20:10 PM
CAS classification : [[_homogeneous, `class D`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.068 (sec). Leaf size: 20

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

\[ y \left (x \right ) = -\frac {x}{2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {x^{2}}{4}} c_{1} x}{2}\right )} \]

✓ Solution by Mathematica

Time used: 3.361 (sec). Leaf size: 37

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

\begin{align*} y(x)\to -\frac {x}{2 W\left (-\frac {1}{2} x e^{\frac {1}{4} \left (x^2-2 c_1\right )}\right )} \\ y(x)\to 0 \\ \end{align*}