23.1.173 problem 173

Internal problem ID [4780]
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 : 173
Date solved : Tuesday, September 30, 2025 at 08:35:12 AM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }&=k +a \,x^{n}+b y+c y^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 231

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

\[ y = \frac {2 \left (\operatorname {BesselY}\left (\frac {\sqrt {b^{2}-4 c k}}{n}+1, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_1 +\operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-4 c k}}{n}+1, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right ) \sqrt {a c}\, x^{\frac {n}{2}}-\left (\operatorname {BesselY}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_1 +\operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right ) \left (\sqrt {b^{2}-4 c k}+b \right )}{2 c \left (\operatorname {BesselY}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_1 +\operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-4 c k}}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )} \]

✓ Mathematica. Time used: 0.405 (sec). Leaf size: 806

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

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