23.1.573 problem 563

Internal problem ID [5180]
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 : 563
Date solved : Tuesday, September 30, 2025 at 11:50:14 AM
CAS classification : [_separable]

\begin{align*} x \left (a +b y\right ) y^{\prime }&=c y \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 44

ode:=x*(a+b*y(x))*diff(y(x),x) = c*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.009 (sec). Leaf size: 36

ode=x*(a+b*y[x])*D[y[x],x]==c*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {a W\left (\frac {b e^{\frac {c_1}{a}} x^{\frac {c}{a}}}{a}\right )}{b}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.241 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \frac {a W\left (\frac {b e^{\frac {C_{1} + c \log {\left (x \right )}}{a}}}{a}\right )}{b} \]