29.10.1 problem 267

Internal problem ID [4867]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 10
Problem number : 267
Date solved : Tuesday, March 04, 2025 at 07:24:42 PM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 220

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

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

✓ Mathematica. Time used: 1.028 (sec). Leaf size: 1981

ode=x^2 D[y[x],x]==a+b x^n+c x^2 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") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a - b*x**n - c*x**2*y(x)**2 + x**2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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