9.1.5 problem problem 42

Internal problem ID [932]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 5.2, Higher-Order Linear Differential Equations. General solutions of Linear Equations. Page 288
Problem number : problem 42
Date solved : Monday, January 27, 2025 at 03:22:35 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([(1-x^2)*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: 0.075 (sec). Leaf size: 39

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

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