90.24.4 problem 4

Internal problem ID [25369]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 5. Second Order Linear Differential Equations. Exercises at page 371
Problem number : 4
Date solved : Friday, October 03, 2025 at 12:00:40 AM
CAS classification : [[_2nd_order, _missing_y]]

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

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 11

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

\[ y = c_1 +\frac {c_2}{t} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 15

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

\begin{align*} y(t)&\to c_2-\frac {c_1}{t} \end{align*}

✓ Sympy. Time used: 0.086 (sec). Leaf size: 7

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

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