23.3.172 problem 174

Internal problem ID [5886]
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 : 174
Date solved : Friday, October 03, 2025 at 01:45:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} a k \,x^{-1+k} y+2 a \,x^{k} y^{\prime }+2 y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.168 (sec). Leaf size: 82

ode:=a*k*x^(-1+k)*y(x)+2*a*x^k*diff(y(x),x)+2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 120

ode=a*k*x^(-1 + k)*y[x] + 2*a*x^k*D[y[x],x] + 2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \left (\frac {1}{k}+1\right )^{-\frac {1}{k+1}} k^{-\frac {1}{k+1}} a^{\frac {1}{k+1}} \left (x^k\right )^{\frac {1}{k}} \operatorname {Hypergeometric1F1}\left (\frac {k+2}{2 k+2},\frac {k+2}{k+1},-\frac {a \left (x^k\right )^{1+\frac {1}{k}}}{k+1}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {k}{2 k+2},\frac {k}{k+1},-\frac {a \left (x^k\right )^{1+\frac {1}{k}}}{k+1}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
k = symbols("k") 
y = Function("y") 
ode = Eq(a*k*x**(k - 1)*y(x) + 2*a*x**k*Derivative(y(x), x) + 2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

RecursionError : maximum recursion depth exceeded