61.30.19 problem 167

Internal problem ID [12667]
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 : Tuesday, January 28, 2025 at 08:10:30 PM
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*}

✓ Solution by Maple

Time used: 1.214 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

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

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

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