1.7 problem 7

Internal problem ID [12424]

Book: DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section: Chapter 8. Differential equations. Exercises page 595
Problem number: 7.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x -y a^{2}=0} \]

Solution by Maple

Time used: 0.0 (sec). Leaf size: 37

dsolve((1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)-a^2*y(x)=0,y(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: 0.151 (sec). Leaf size: 89

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

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