23.3.200 problem 202

Internal problem ID [5914]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 202
Date solved : Tuesday, September 30, 2025 at 02:06:08 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 71

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

\[ y = \left (\operatorname {BesselY}\left (\frac {a -1}{k +1}, \frac {2 \sqrt {b}\, x^{\frac {k}{2}+\frac {1}{2}}}{k +1}\right ) c_2 +\operatorname {BesselJ}\left (\frac {a -1}{k +1}, \frac {2 \sqrt {b}\, x^{\frac {k}{2}+\frac {1}{2}}}{k +1}\right ) c_1 \right ) x^{-\frac {a}{2}+\frac {1}{2}} \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 165

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

\begin{align*} y(x)&\to \left (\frac {1}{k}+1\right )^{\frac {a-1}{k+1}} k^{\frac {a-1}{k+1}} b^{\frac {1-a}{2 k+2}} \left (x^k\right )^{-\frac {a-1}{2 k}} \left (c_2 \operatorname {Gamma}\left (\frac {-a+k+2}{k+1}\right ) \operatorname {BesselJ}\left (\frac {1-a}{k+1},\frac {2 \sqrt {b} \left (x^k\right )^{\frac {k+1}{2 k}}}{k+1}\right )+c_1 \operatorname {Gamma}\left (\frac {a+k}{k+1}\right ) \operatorname {BesselJ}\left (\frac {a-1}{k+1},\frac {2 \sqrt {b} \left (x^k\right )^{\frac {k+1}{2 k}}}{k+1}\right )\right ) \end{align*}

✗ Sympy

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

TypeError : invalid input: 1 - a