76.11.11 problem 25

Internal problem ID [17476]
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 4.1 (Definitions and examples). Problems at page 214
Problem number : 25
Date solved : Thursday, March 13, 2025 at 10:09:24 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+4 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: 0.051 (sec). Leaf size: 32

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+4*y(t) = 0; 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\frac {3 t}{2}} \left (3 \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right )+7 \cos \left (\frac {\sqrt {7}\, t}{2}\right )\right )}{7} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 48

ode=D[y[t],{t,2}]+3*D[y[t],t]+4*y[t]==0; 
ic={y[0]==1,Derivative[1][y][0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{7} e^{-3 t/2} \left (3 \sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right )+7 \cos \left (\frac {\sqrt {7} t}{2}\right )\right ) \]

✓ Sympy. Time used: 0.198 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + 3*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)
 

\[ y{\left (t \right )} = \left (\frac {3 \sqrt {7} \sin {\left (\frac {\sqrt {7} t}{2} \right )}}{7} + \cos {\left (\frac {\sqrt {7} t}{2} \right )}\right ) e^{- \frac {3 t}{2}} \]