13.13.9 problem 9

Internal problem ID [2439]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8.1, Singular points, Euler equations. Page 201
Problem number : 9
Date solved : Monday, January 27, 2025 at 05:52:51 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: 0.011 (sec). Leaf size: 9

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)
 

\[ y = \sin \left (\ln \left (t \right )\right ) t \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 10

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]
 

\[ y(t)\to t \sin (\log (t)) \]