60.3.331 problem 1348

Internal problem ID [11327]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1348
Date solved : Sunday, March 30, 2025 at 08:16:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.100 (sec). Leaf size: 73

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

\[ y = \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{x^{2}-1}\right ) \left (c_1 +c_2 \int \frac {1}{x \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{x^{2}-1}\right )^{2}}d x \right ) \]

✓ Mathematica. Time used: 0.49 (sec). Leaf size: 34

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

\[ y(x)\to c_1 \text {MathieuC}[-b,a,i \log (x)]+c_2 \text {MathieuS}[-b,a,i \log (x)] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + Derivative(y(x), x)/x + (a*(x**4 + 1) + b*x**2)*y(x)/x**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a*x*y(x) + a*y(x)/x**3 + b*y(x)/x + x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) cannot be solved by the factorable group method