23.3.258 problem 260

Internal problem ID [5972]
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 : 260
Date solved : Friday, October 03, 2025 at 01:45:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{k} \left (a +b \,x^{k}\right ) y+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.121 (sec). Leaf size: 74

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

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

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 147

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

\begin{align*} y(x)&\to 2^{\frac {k+1}{2 k}} x^{\frac {1}{2}-\frac {k}{2}} \left (x^k\right )^{\frac {k+1}{2 k}} e^{\frac {i \sqrt {b} x^k}{k}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {-\frac {i a}{\sqrt {b}}+k+1}{2 k},1+\frac {1}{k},-\frac {2 i \sqrt {b} x^k}{k}\right )+c_2 L_{-\frac {-\frac {i a}{\sqrt {b}}+k+1}{2 k}}^{\frac {1}{k}}\left (-\frac {2 i \sqrt {b} x^k}{k}\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(x**2*Derivative(y(x), (x, 2)) + x**k*(a + b*x**k)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False