23.1.588 problem 581

Internal problem ID [5195]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 581
Date solved : Tuesday, September 30, 2025 at 11:52:50 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class C`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 59

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

\begin{align*} y &= \frac {-2+\sqrt {4-2 \ln \left (x \right )+2 c_1}}{2 \left (\ln \left (x \right )-c_1 \right ) x} \\ y &= \frac {2+\sqrt {4-2 \ln \left (x \right )+2 c_1}}{2 \left (-\ln \left (x \right )+c_1 \right ) x} \\ \end{align*}

✓ Mathematica. Time used: 0.462 (sec). Leaf size: 79

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

\begin{align*} y(x)&\to \frac {x}{-2 x^2+\frac {\sqrt {x (-2 \log (x)+4+c_1)}}{\sqrt {\frac {1}{x^3}}}}\\ y(x)&\to -\frac {x}{2 x^2+\frac {\sqrt {x (-2 \log (x)+4+c_1)}}{\sqrt {\frac {1}{x^3}}}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.814 (sec). Leaf size: 56

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

\[ \left [ y{\left (x \right )} = \frac {- \frac {\sqrt {2} \sqrt {- C_{1} - \log {\left (x \right )} + 2}}{2} - 1}{x \left (C_{1} + \log {\left (x \right )}\right )}, \ y{\left (x \right )} = \frac {\frac {\sqrt {2} \sqrt {- C_{1} - \log {\left (x \right )} + 2}}{2} - 1}{x \left (C_{1} + \log {\left (x \right )}\right )}\right ] \]