61.30.19 problem 167

Internal problem ID [12588]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 167
Date solved : Monday, March 31, 2025 at 05:52:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.124 (sec). Leaf size: 27

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

\[ y = c_1 \operatorname {MathieuC}\left (a +b , -\frac {a}{2}, \arccos \left (x \right )\right )+c_2 \operatorname {MathieuS}\left (a +b , -\frac {a}{2}, \arccos \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 34

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

\[ y(x)\to c_1 \text {MathieuC}\left [a+b,-\frac {a}{2},\arccos (x)\right ]+c_2 \text {MathieuS}\left [a+b,-\frac {a}{2},\arccos (x)\right ] \]

✗ Sympy

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

False