9.9 problem 12

Internal problem ID [1747]

Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.2.2, Equal roots, reduction of order. Page 147
Problem number: 12.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [_Gegenbauer]

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

Solution by Maple

Time used: 0.008 (sec). Leaf size: 26

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

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

Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 19

DSolve[(1-t^2)*y''[t]-2*t*y'[t]+2*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to c_1 t+c_2 \left (t \tanh ^{-1}(t)-1\right ) \\ \end{align*}