13.8.11 problem 19

Internal problem ID [2386]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.1, Complex roots. Page 141
Problem number : 19
Date solved : Tuesday, March 04, 2025 at 02:09:11 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 29

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

\[ y = \frac {c_1 \sin \left (\frac {\sqrt {7}\, \ln \left (t \right )}{2}\right )+c_2 \cos \left (\frac {\sqrt {7}\, \ln \left (t \right )}{2}\right )}{\sqrt {t}} \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 42

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

\[ y(t)\to \frac {c_2 \cos \left (\frac {1}{2} \sqrt {7} \log (t)\right )+c_1 \sin \left (\frac {1}{2} \sqrt {7} \log (t)\right )}{\sqrt {t}} \]

✓ Sympy. Time used: 0.226 (sec). Leaf size: 34

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

\[ y{\left (t \right )} = \frac {C_{1} \sin {\left (\frac {\sqrt {7} \log {\left (t \right )}}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {7} \log {\left (t \right )}}{2} \right )}}{\sqrt {t}} \]