74.10.26 problem 26

Internal problem ID [16160]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.2, page 147
Problem number : 26
Date solved : Monday, March 31, 2025 at 02:45:36 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.079 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = 0; 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = {\mathrm e}^{-t} \left (\frac {\sin \left (2 t \right )}{2}+\cos \left (2 t \right )\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 25

ode=D[y[t],{t,2}]+2*D[y[t],t]+5*y[t]==0; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{2} e^{-t} (\sin (2 t)+2 \cos (2 t)) \]

✓ Sympy. Time used: 0.166 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {\sin {\left (2 t \right )}}{2} + \cos {\left (2 t \right )}\right ) e^{- t} \]