23.1.336 problem 322

Internal problem ID [4943]
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 : 322
Date solved : Tuesday, September 30, 2025 at 09:03:34 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.263 (sec). Leaf size: 64

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] (b+c K[1])}dK[1]\&\right ]\left [\int _1^x\frac {1}{K[2]^2+a K[2]}dK[2]+c_1\right ]\\ y(x)&\to 0\\ y(x)&\to -\frac {b}{c} \end{align*}

✓ Sympy. Time used: 2.782 (sec). Leaf size: 48

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

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