83.38.4 problem 4

Internal problem ID [19463]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (C) at page 133
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 01:45:13 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 }+m^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 45

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

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