7.18.7 problem 7

Internal problem ID [536]
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 : 7
Date solved : Tuesday, March 04, 2025 at 11:26:19 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+x&=\cos \left (3 t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.231 (sec). Leaf size: 15

ode:=diff(diff(x(t),t),t)+x(t) = cos(3*t); 
ic:=x(0) = 1, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 20

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

\[ x(t)\to \frac {1}{8} (9 \cos (t)-\cos (3 t)) \]

✓ Sympy. Time used: 0.083 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) - cos(3*t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {9 \cos {\left (t \right )}}{8} - \frac {\cos {\left (3 t \right )}}{8} \]