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 : Sunday, March 30, 2025 at 12:01:28 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*}

✓ Maple. Time used: 0.058 (sec). Leaf size: 9

ode:=t^2*diff(diff(y(t),t),t)-t*diff(y(t),t)+2*y(t) = 0; 
ic:=y(1) = 0, D(y)(1) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

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

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

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

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

✓ Sympy. Time used: 0.202 (sec). Leaf size: 8

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) - t*Derivative(y(t), t) + 2*y(t),0) 
ics = {y(1): 0, Subs(Derivative(y(t), t), t, 1): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = t \sin {\left (\log {\left (t \right )} \right )} \]