76.12.19 problem 31

Internal problem ID [17504]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 31
Date solved : Monday, March 31, 2025 at 04:16:02 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} t^{2} y^{\prime \prime }+3 t y^{\prime }+y&=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: 14

ode:=t^2*diff(diff(y(t),t),t)+3*t*diff(y(t),t)+y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 17

ode=t^2*D[y[t],{t,2}]+3*t*D[y[t],t]+y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {c_2 \log (t)+c_1}{t} \]

✓ Sympy. Time used: 0.148 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + 3*t*Derivative(y(t), t) + y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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