76.18.2 problem 13

Internal problem ID [17642]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 13
Date solved : Monday, March 31, 2025 at 04:23:27 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 9 y^{\prime \prime }+12 y^{\prime }+4 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Maple. Time used: 0.104 (sec). Leaf size: 13

ode:=9*diff(diff(y(t),t),t)+12*diff(y(t),t)+4*y(t) = 0; 
ic:=y(0) = 2, D(y)(0) = -1; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{-\frac {2 t}{3}} \left (6+t \right )}{3} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 19

ode=9*D[y[t],{t,2}]+12*D[y[t],t]+4*y[t]==0; 
ic={y[0]==2,Derivative[1][y][0] == -1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{3} e^{-2 t/3} (t+6) \]

✓ Sympy. Time used: 0.160 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) + 12*Derivative(y(t), t) + 9*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics)
 

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