14.13.9 problem 9

Internal problem ID [2636]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8.1, singular points, Euler equations. Excercises page 203
Problem number : 9
Date solved : Monday, January 27, 2025 at 06:05:04 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t^{2} y^{\prime \prime }-t y^{\prime }-2 y&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 25

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

\[ y = \frac {\sqrt {3}\, t \left (t^{\sqrt {3}}-t^{-\sqrt {3}}\right )}{6} \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 36

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

\[ y(t)\to \frac {t^{1-\sqrt {3}} \left (t^{2 \sqrt {3}}-1\right )}{2 \sqrt {3}} \]