60.3.200 problem 1214

Internal problem ID [11196]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1214
Date solved : Thursday, March 13, 2025 at 08:25:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (-x^{4}+\left (2 n +2 a +1\right ) x^{2}+\left (-1\right )^{n} a -a^{2}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.377 (sec). Leaf size: 71

ode:=x^2*diff(diff(y(x),x),x)+(-x^4+(2*n+2*a+1)*x^2+a*(-1)^n-a^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\operatorname {WhittakerM}\left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right ) c_{1} +\operatorname {WhittakerW}\left (\frac {a}{2}+\frac {n}{2}+\frac {1}{4}, \frac {\sqrt {1-4 a \left (-1\right )^{n}+4 a^{2}}}{4}, x^{2}\right ) c_{2}}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.366 (sec). Leaf size: 191

ode=((-1)^n*a - a^2 + (1 + 2*a + 2*n)*x^2 - x^4)*y[x] + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{-\frac {x^2}{2}} 2^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} \left (x^2\right )^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{4} \left (-2 a-2 n+\sqrt {4 a^2-4 (-1)^n a+1}+1\right ),\frac {1}{2} \left (\sqrt {4 a^2-4 (-1)^n a+1}+2\right ),x^2\right )+c_2 L_{\frac {1}{4} \left (2 a+2 n-\sqrt {4 a^2-4 (-1)^n a+1}-1\right )}^{\frac {1}{2} \sqrt {4 a^2-4 (-1)^n a+1}}\left (x^2\right )\right )}{\sqrt {x}} \]

✗ 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)) + ((-1)**n*a - a**2 - x**4 + x**2*(2*a + 2*n + 1))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None