56.5.3 problem Ex 3

Internal problem ID [14099]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 12. Page 18
Problem number : Ex 3
Date solved : Thursday, October 02, 2025 at 09:13:35 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 5.083 (sec). Leaf size: 35

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

\begin{align*} y(x)&\to -\frac {1}{x W\left (\frac {e^{-1+\frac {9 c_1}{2^{2/3}}}}{x^2}\right )}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.528 (sec). Leaf size: 19

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

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