23.1.181 problem 181

Internal problem ID [4788]
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 : 181
Date solved : Tuesday, September 30, 2025 at 08:36:11 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} x y^{\prime }+b x +\left (2+a x y\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 80

ode:=x*diff(y(x),x)+b*x+(2+a*x*y(x))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-2 a b c_1 x -i {\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x} \sqrt {a}\, \sqrt {b}\, x -2 i c_1 \sqrt {a}\, \sqrt {b}-{\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x}}{x a \left (2 i c_1 \sqrt {a}\, \sqrt {b}+{\mathrm e}^{-2 i \sqrt {a}\, \sqrt {b}\, x}\right )} \]

✓ Mathematica. Time used: 1.776 (sec). Leaf size: 43

ode=x*D[y[x],x]+b*x+(2+a*x*y[x])*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{a x}-\sqrt {\frac {b}{a}} \tan \left (a x \sqrt {\frac {b}{a}}-c_1\right ) \end{align*}

✗ Sympy

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

NonlinearError : nonlinear term: sqrt(-a*b)