76.11.3 problem 3

Internal problem ID [17547]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.1 (Definitions and examples). Problems at page 214
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 08:27:43 PM
CAS classification : [_Gegenbauer]

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

✓ Solution by Maple

Time used: 0.271 (sec). Leaf size: 15

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

\[ y = c_{1} \operatorname {LegendreP}\left (\alpha , x\right )+c_{2} \operatorname {LegendreQ}\left (\alpha , x\right ) \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 18

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

\[ y(x)\to c_1 \operatorname {LegendreP}(a,x)+c_2 \operatorname {LegendreQ}(a,x) \]