47.2.14 problem 14

Internal problem ID [7430]
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 : 14
Date solved : Sunday, March 30, 2025 at 12:03:22 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 17

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

\[ y = -\frac {x}{\operatorname {LambertW}\left (-x \,{\mathrm e}^{-c_1}\right )} \]

✓ Mathematica. Time used: 4.072 (sec). Leaf size: 27

ode=x*D[y[x],x]-y[x]==y[x]*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.658 (sec). Leaf size: 14

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

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