49.15.11 problem 7

Internal problem ID [7697]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 130
Problem number : 7
Date solved : Monday, January 27, 2025 at 03:09:59 PM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 33

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

\[ y = c_{1} \left (x +\sqrt {x^{2}-1}\right )^{-\alpha }+c_{2} \left (x +\sqrt {x^{2}-1}\right )^{\alpha } \]

✓ Solution by Mathematica

Time used: 0.051 (sec). Leaf size: 45

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

\[ y(x)\to c_1 \cosh \left (\alpha \log \left (\sqrt {x^2-1}+x\right )\right )+i c_2 \sinh \left (\alpha \log \left (\sqrt {x^2-1}+x\right )\right ) \]