23.1.598 problem 591

Internal problem ID [5205]
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 : 591
Date solved : Tuesday, September 30, 2025 at 11:55:01 AM
CAS classification : [_separable]

\begin{align*} x y \left (b \,x^{2}+a \right ) y^{\prime }&=A +B y^{2} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 82

ode:=x*y(x)*(b*x^2+a)*diff(y(x),x) = A+B*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {-B \left (-x^{\frac {2 B}{a}} \left (b \,x^{2}+a \right )^{-\frac {B}{a}} c_1 B +A \right )}}{B} \\ y &= -\frac {\sqrt {-B \left (-x^{\frac {2 B}{a}} \left (b \,x^{2}+a \right )^{-\frac {B}{a}} c_1 B +A \right )}}{B} \\ \end{align*}

✓ Mathematica. Time used: 1.144 (sec). Leaf size: 130

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

\begin{align*} y(x)&\to -\frac {\sqrt {-A+\exp \left (2 B \left (\int _1^x\frac {1}{b K[1]^3+a K[1]}dK[1]+c_1\right )\right )}}{\sqrt {B}}\\ y(x)&\to \frac {\sqrt {-A+\exp \left (2 B \left (\int _1^x\frac {1}{b K[1]^3+a K[1]}dK[1]+c_1\right )\right )}}{\sqrt {B}}\\ y(x)&\to -\frac {i \sqrt {A}}{\sqrt {B}}\\ y(x)&\to \frac {i \sqrt {A}}{\sqrt {B}} \end{align*}

✓ Sympy. Time used: 1.814 (sec). Leaf size: 63

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

\[ \left [ y{\left (x \right )} = \sqrt {\frac {- A + e^{B \left (C_{1} + \frac {2 \log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{a}\right )}}{B}}, \ y{\left (x \right )} = - \sqrt {\frac {- A + e^{B \left (C_{1} + \frac {2 \log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{a}\right )}}{B}}\right ] \]