90.31.11 problem 13

Internal problem ID [25461]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 7. Power series methods. Exercises at page 537
Problem number : 13
Date solved : Friday, October 03, 2025 at 12:01:45 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t^{2} y^{\prime \prime }+5 t y^{\prime }+4 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {1}{t^{2}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

ode:=t^2*diff(diff(y(t),t),t)+5*t*diff(y(t),t)+4*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 18

ode=t^2*D[y[t],{t,2}]+5*t*D[y[t],t]+4*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {2 c_2 \log (t)+c_1}{t^2} \end{align*}

✓ Sympy. Time used: 0.105 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + 5*t*Derivative(y(t), t) + 4*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {C_{1} + C_{2} \log {\left (t \right )}}{t^{2}} \]