1.18.6 problem 6

Internal problem ID [535]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.2 (Transformation of initial value problems). Problems at page 287
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 04:01:05 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x&=\cos \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.169 (sec). Leaf size: 16

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

\[ x = \frac {\cos \left (t \right )}{3}-\frac {2 \cos \left (t \right )^{2}}{3}+\frac {1}{3} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 23

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

\begin{align*} x(t)&\to \frac {2}{3} \sin ^2\left (\frac {t}{2}\right ) (2 \cos (t)+1) \end{align*}

✓ Sympy. Time used: 0.049 (sec). Leaf size: 14

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

\[ x{\left (t \right )} = \frac {\cos {\left (t \right )}}{3} - \frac {\cos {\left (2 t \right )}}{3} \]