23.1.87 problem 81

Internal problem ID [4694]
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 : 81
Date solved : Tuesday, September 30, 2025 at 08:15:12 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 70

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

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

✓ Mathematica. Time used: 0.118 (sec). Leaf size: 75

ode=D[y[x],x]==y[x]*(a+b*y[x]^2); 
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] \left (b K[1]^2+a\right )}dK[1]\&\right ][x+c_1]\\ y(x)&\to 0\\ y(x)&\to -\frac {i \sqrt {a}}{\sqrt {b}}\\ y(x)&\to \frac {i \sqrt {a}}{\sqrt {b}} \end{align*}

✓ Sympy. Time used: 4.198 (sec). Leaf size: 58

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

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