23.3.131 problem 133

Internal problem ID [5845]
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 : 133
Date solved : Tuesday, September 30, 2025 at 02:04:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 209

ode:=a*k*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);
 

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

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 74

ode=a*k*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 e^{-\frac {a x^{k+1}}{k+1}} \left (c_2-\frac {c_1 x \left (-\frac {a x^{k+1}}{k+1}\right )^{-\frac {1}{k+1}} \Gamma \left (\frac {1}{k+1},-\frac {a x^{k+1}}{k+1}\right )}{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) + a*x**k*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

RecursionError : maximum recursion depth exceeded