1.19.1 problem 27

Internal problem ID [541]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.3 (Translation and partial fractions). Problems at page 296
Problem number : 27
Date solved : Tuesday, September 30, 2025 at 04:01:09 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+6 x^{\prime }+25 x&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} x \left (0\right )&=2 \\ x^{\prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 23

ode:=diff(diff(x(t),t),t)+6*diff(x(t),t)+25*x(t) = 0; 
ic:=[x(0) = 2, D(x)(0) = 3]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = \frac {{\mathrm e}^{-3 t} \left (8 \cos \left (4 t \right )+9 \sin \left (4 t \right )\right )}{4} \]

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

ode=D[x[t],{t,2}]+6*D[x[t],t]+25*x[t]==0; 
ic={x[0]==2,Derivative[1][x][0] ==3}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} e^{-3 t} (9 \sin (4 t)+8 \cos (4 t)) \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(25*x(t) + 6*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 2, Subs(Derivative(x(t), t), t, 0): 3} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {9 \sin {\left (4 t \right )}}{4} + 2 \cos {\left (4 t \right )}\right ) e^{- 3 t} \]