61.3.10 problem Problem 11

Internal problem ID [15309]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.5 Laplace transform. Homogeneous equations. Problems page 357
Problem number : Problem 11
Date solved : Thursday, October 02, 2025 at 10:11:31 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+13 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=-6 \\ \end{align*}

✓ Maple. Time used: 0.127 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+13*y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = -6]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {{\mathrm e}^{-2 t} \left (3 \cos \left (3 t \right )-4 \sin \left (3 t \right )\right )}{3} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 27

ode=D[y[t],{t,2}]+4*D[y[t],t]+13*y[t]==0; 
ic={y[0]==1,Derivative[1][y][0] ==-6}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{3} e^{-2 t} (3 \cos (3 t)-4 \sin (3 t)) \end{align*}

✓ Sympy. Time used: 0.115 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(13*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -6} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {4 \sin {\left (3 t \right )}}{3} + \cos {\left (3 t \right )}\right ) e^{- 2 t} \]