23.3.277 problem 279

Internal problem ID [5991]
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 : 279
Date solved : Friday, October 03, 2025 at 01:45:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -\left (-x^{4}+4 a \,x^{2}+n^{2}\right ) y+x y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 41

ode:=-(-x^4+4*a*x^2+n^2)*y(x)+x*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.113 (sec). Leaf size: 93

ode=-((n^2 + 4*a*x^2 - x^4)*y[x]) + x*D[y[x],x] + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2^{\frac {n+1}{2}} e^{-\frac {i x^2}{2}} \left (x^2\right )^{\frac {n+1}{2}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} (-2 i a+n+1),n+1,i x^2\right )+c_2 L_{i a-\frac {n}{2}-\frac {1}{2}}^n\left (i x^2\right )\right )}{x} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (-4*a*x**2 - n**2 + x**4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None