14.9.9 problem 12

Internal problem ID [2575]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2.2. Equal roots, reduction of order. Excercises page 149
Problem number : 12
Date solved : Monday, January 27, 2025 at 06:01:08 AM
CAS classification : [_Gegenbauer]

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

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 25

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

\[ y = \frac {c_2 \ln \left (-1+t \right ) t}{2}-\frac {c_2 \ln \left (t +1\right ) t}{2}+c_1 t +c_2 \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 33

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

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