41.2.48 problem 44

Internal problem ID [8750]
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 : 44
Date solved : Tuesday, September 30, 2025 at 05:50:10 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 18

ode:=y(x)+x*(2*x*y(x)+1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {1}{2 \operatorname {LambertW}\left (\frac {c_1}{2 x}\right ) x} \]

✓ Mathematica. Time used: 0.082 (sec). Leaf size: 72

ode=y[x]+x*(2*x*y[x]+1)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {(-1)^{2/3} (1-4 x y(x))}{\sqrt [3]{2} (2 x y(x)+1)}}\frac {2}{2 K[1]^3+3 \sqrt [3]{-2} K[1]+2}dK[1]+\frac {1}{9} (-2)^{2/3} \log (x)=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.407 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(2*x*y(x) + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = e^{C_{1} + W\left (\frac {e^{- C_{1}}}{2 x}\right )} \]