68.9.21 problem 32

Internal problem ID [17484]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.1, page 141
Problem number : 32
Date solved : Thursday, October 02, 2025 at 02:24:38 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t^{2} y^{\prime \prime }+6 t y^{\prime }+6 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: 0.004 (sec). Leaf size: 13

ode:=t^2*diff(diff(y(t),t),t)+6*t*diff(y(t),t)+6*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {c_2 t +c_1}{t^{3}} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 16

ode=t^2*D[y[t],{t,2}]+6*t*D[y[t],t]+6*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {c_2 t+c_1}{t^3} \end{align*}

✓ Sympy. Time used: 0.099 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + 6*t*Derivative(y(t), t) + 6*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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