83.41.8 problem 2 (vii)

Internal problem ID [19488]
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 at end of chapter VIII. Page 141
Problem number : 2 (vii)
Date solved : Tuesday, January 28, 2025 at 01:46:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 19

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

\[ y \left (x \right ) = c_{1} \sin \left (a \,\operatorname {arctanh}\left (x \right )\right )+c_{2} \cos \left (a \,\operatorname {arctanh}\left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 2.043 (sec). Leaf size: 50

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

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