74.1.5 problem 6

Internal problem ID [15714]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 6
Date solved : Monday, March 31, 2025 at 01:45:51 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 23

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

\[ y = c_1 \sin \left (\sqrt {2}\, \ln \left (t \right )\right )+c_2 \cos \left (\sqrt {2}\, \ln \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 30

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

\[ y(t)\to c_1 \cos \left (\sqrt {2} \log (t)\right )+c_2 \sin \left (\sqrt {2} \log (t)\right ) \]

✓ Sympy. Time used: 0.189 (sec). Leaf size: 26

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

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