74.10.45 problem 43

Internal problem ID [16179]
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 : 43
Date solved : Monday, March 31, 2025 at 02:46:05 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+3 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.054 (sec). Leaf size: 29

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

\[ y = \frac {\left (-b -a \right ) {\mathrm e}^{-3 t}}{2}+\frac {3 \,{\mathrm e}^{-t} \left (a +\frac {b}{3}\right )}{2} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 35

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

\[ y(t)\to \frac {1}{2} e^{-3 t} \left (a \left (3 e^{2 t}-1\right )+b \left (e^{2 t}-1\right )\right ) \]

✓ Sympy. Time used: 0.188 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(3*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): a, Subs(Derivative(y(t), t), t, 0): b} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {3 a}{2} + \frac {b}{2} + \left (- \frac {a}{2} - \frac {b}{2}\right ) e^{- 2 t}\right ) e^{- t} \]