64.9.3 problem 3

Internal problem ID [13401]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.1. Basic theory of linear differential equations. Exercises page 124
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 05:41:47 AM
CAS classification : [_Gegenbauer]

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

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 14

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

\[ y = c_{2} x^{2}+c_{1} x +c_{2} \]

✓ Solution by Mathematica

Time used: 0.377 (sec). Leaf size: 75

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

\[ y(x)\to \sqrt {x^2-1} \exp \left (\int _1^x\frac {K[1]+2}{K[1]^2-1}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+2}{K[1]^2-1}dK[1]\right )dK[2]+c_1\right ) \]