61.30.25 problem 173

Internal problem ID [12594]
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 : 173
Date solved : Monday, March 31, 2025 at 05:52:43 AM
CAS classification : [_Jacobi]

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

✓ Maple. Time used: 0.055 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.152 (sec). Leaf size: 50

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

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

✗ Sympy

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

False