70.18.4 problem 15

Internal problem ID [19001]
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 : 15
Date solved : Thursday, October 02, 2025 at 03:36:52 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 6 y^{\prime \prime }+5 y^{\prime }+y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=4 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 17

ode:=6*diff(diff(y(t),t),t)+5*diff(y(t),t)+y(t) = 0; 
ic:=[y(0) = 4, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 12 \,{\mathrm e}^{-\frac {t}{3}}-8 \,{\mathrm e}^{-\frac {t}{2}} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 25

ode=6*D[y[t],{t,2}]+5*D[y[t],t]+y[t]==0; 
ic={y[0]==4,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 4 e^{-t/2} \left (3 e^{t/6}-2\right ) \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) + 5*Derivative(y(t), t) + 6*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 4, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - 8 e^{- \frac {t}{2}} + 12 e^{- \frac {t}{3}} \]