82.54.12 problem Ex. 12

Internal problem ID [19028]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at end of chapter at page 120
Problem number : Ex. 12
Date solved : Tuesday, January 28, 2025 at 12:46:04 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^{2} y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{a \arcsin \left (x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 35

dsolve([(1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)-a^2*y(x)=0,exp(a*arcsin(x))],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 42

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

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