7.19.12 problem 38

Internal problem ID [552]
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 : 38
Date solved : Tuesday, March 04, 2025 at 11:26:34 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+6 x^{\prime }+18 x&=\cos \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.223 (sec). Leaf size: 36

ode:=diff(diff(x(t),t),t)+6*diff(x(t),t)+18*x(t) = cos(2*t); 
ic:=x(0) = 1, D(x)(0) = -1; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x \left (t \right ) = \frac {7 \cos \left (2 t \right )}{170}+\frac {3 \sin \left (2 t \right )}{85}+\frac {{\mathrm e}^{-3 t} \left (489 \cos \left (3 t \right )+307 \sin \left (3 t \right )\right )}{510} \]

✓ Mathematica. Time used: 0.203 (sec). Leaf size: 49

ode=D[x[t],{t,2}]+6*D[x[t],t]+18*x[t]==Cos[2*t]; 
ic={x[0]==1,Derivative[1][x][0] ==-1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {1}{510} e^{-3 t} \left (18 e^{3 t} \sin (2 t)+307 \sin (3 t)+21 e^{3 t} \cos (2 t)+489 \cos (3 t)\right ) \]

✓ Sympy. Time used: 0.262 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(18*x(t) - cos(2*t) + 6*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): -1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {307 \sin {\left (3 t \right )}}{510} + \frac {163 \cos {\left (3 t \right )}}{170}\right ) e^{- 3 t} + \frac {3 \sin {\left (2 t \right )}}{85} + \frac {7 \cos {\left (2 t \right )}}{170} \]