61.30.25 problem 173

Internal problem ID [12673]
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 : Tuesday, January 28, 2025 at 08:10:42 PM
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*}

✓ Solution by Maple

Time used: 1.159 (sec). Leaf size: 39

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

\[ y = c_{1} \operatorname {MathieuC}\left (-a -2 b , \frac {a}{2}, \arccos \left (\sqrt {x}\right )\right )+c_{2} \operatorname {MathieuS}\left (-a -2 b , \frac {a}{2}, \arccos \left (\sqrt {x}\right )\right ) \]

✓ Solution by Mathematica

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

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

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