60.3.230 problem 1235

Internal problem ID [11240]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1235
Date solved : Monday, January 27, 2025 at 10:49:42 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 }+a y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 50

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

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