68.9.20 problem 31

Internal problem ID [17483]
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 : 31
Date solved : Thursday, October 02, 2025 at 02:24:37 PM
CAS classification : [[_Emden, _Fowler]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {1}{t^{4}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 15

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

\[ y = \frac {c_2 \,t^{5}+c_1}{t^{4}} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 16

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

\begin{align*} y(t)&\to \frac {c_1}{t^4}+c_2 t \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 10

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

\[ y{\left (t \right )} = \frac {C_{1}}{t^{4}} + C_{2} t \]