23.3.132 problem 134

Internal problem ID [5846]
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 : 134
Date solved : Tuesday, September 30, 2025 at 02:04:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.096 (sec). Leaf size: 122

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

\[ y = c_1 x +\frac {c_2 \,x^{-\frac {3 k}{2}} \left (\left (k +1\right ) \left (a \,x^{k} x +k \right ) \operatorname {WhittakerM}\left (\frac {-k -2}{2+2 k}, \frac {2 k +1}{2+2 k}, \frac {x a \,x^{k}}{k +1}\right )+k^{2} \operatorname {WhittakerM}\left (\frac {k}{2+2 k}, \frac {2 k +1}{2+2 k}, \frac {x a \,x^{k}}{k +1}\right )\right ) {\mathrm e}^{-\frac {a \,x^{k} x}{2+2 k}}}{x} \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 90

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

\begin{align*} y(x)&\to \frac {\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}} \left (c_2 (-1)^{\frac {1}{k+1}} (k+1)-c_1 \Gamma \left (-\frac {1}{k+1},0,\frac {a \left (x^k\right )^{1+\frac {1}{k}}}{k+1}\right )\right )}{k+1} \end{align*}

✗ Sympy

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

RecursionError : maximum recursion depth exceeded