23.1.174 problem 174

Internal problem ID [4781]
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 : 174
Date solved : Tuesday, September 30, 2025 at 08:35:56 AM
CAS classification : [_rational, [_Riccati, _special]]

\begin{align*} x y^{\prime }+a +x y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 59

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

\[ y = \frac {\sqrt {a}\, \left (\operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right ) c_1 +\operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right )\right )}{\sqrt {x}\, \left (c_1 \operatorname {BesselJ}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )+\operatorname {BesselY}\left (1, 2 \sqrt {a}\, \sqrt {x}\right )\right )} \]

✓ Mathematica. Time used: 0.148 (sec). Leaf size: 289

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

\begin{align*} y(x)&\to \frac {2 \sqrt {a} \sqrt {x} \operatorname {BesselY}\left (0,2 \sqrt {a} \sqrt {x}\right )+2 \operatorname {BesselY}\left (1,2 \sqrt {a} \sqrt {x}\right )-2 \sqrt {a} \sqrt {x} \operatorname {BesselY}\left (2,2 \sqrt {a} \sqrt {x}\right )-i \sqrt {a} c_1 \sqrt {x} \operatorname {BesselJ}\left (0,2 \sqrt {a} \sqrt {x}\right )-i c_1 \operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )+i \sqrt {a} c_1 \sqrt {x} \operatorname {BesselJ}\left (2,2 \sqrt {a} \sqrt {x}\right )}{4 x \operatorname {BesselY}\left (1,2 \sqrt {a} \sqrt {x}\right )-2 i c_1 x \operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )}\\ y(x)&\to \frac {\sqrt {a} \sqrt {x} \operatorname {BesselJ}\left (0,2 \sqrt {a} \sqrt {x}\right )+\operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )-\sqrt {a} \sqrt {x} \operatorname {BesselJ}\left (2,2 \sqrt {a} \sqrt {x}\right )}{2 x \operatorname {BesselJ}\left (1,2 \sqrt {a} \sqrt {x}\right )} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE a/x + y(x)**2 + Derivative(y(x), x) cannot be solved by the factorable group method