58.2.17 problem 18

Internal problem ID [9140]
Book : Second order enumerated odes
Section : section 2
Problem number : 18
Date solved : Monday, January 27, 2025 at 05:48:47 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 }-c^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 42

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

\[ y(x)\to c_1 \cos \left (c \log \left (\sqrt {x^2-1}+x\right )\right )+c_2 \sin \left (c \log \left (\sqrt {x^2-1}+x\right )\right ) \]