50.12.5 problem 4

Internal problem ID [8009]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.4. THE USE OF A KNOWN SOLUTION TO FIND ANOTHER. Page 74
Problem number : 4
Date solved : Monday, January 27, 2025 at 03:36:45 PM
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: 0.006 (sec). Leaf size: 25

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

\[ y = -\frac {c_{2} \ln \left (x +1\right ) x}{2}+\frac {c_{2} \ln \left (x -1\right ) x}{2}+c_{1} x +c_{2} \]

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 33

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

\[ y(x)\to c_1 x-\frac {1}{2} c_2 (x \log (1-x)-x \log (x+1)+2) \]