7.19.14 problem 40

Internal problem ID [554]
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 : 40
Date solved : Tuesday, March 04, 2025 at 11:26:36 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+\frac {2 x^{\prime }}{5}+\frac {226 x}{25}&=6 \,{\mathrm e}^{-\frac {t}{5}} \cos \left (3 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: 0.264 (sec). Leaf size: 14

ode:=diff(diff(x(t),t),t)+2/5*diff(x(t),t)+226/25*x(t) = 6*exp(-1/5*t)*cos(3*t); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{5}} \sin \left (3 t \right ) t \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 24

ode=D[x[t],{t,2}]+4/10*D[x[t],t]+904/100*x[t]==6*Exp[-t/5]*Cos[3*t]; 
ic={x[0]==0,Derivative[1][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to e^{-t/5} t \sin (t) (2 \cos (2 t)+1) \]

✓ Sympy. Time used: 0.378 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(226*x(t)/25 + 2*Derivative(x(t), t)/5 + Derivative(x(t), (t, 2)) - 6*exp(-t/5)*cos(3*t),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = t e^{- \frac {t}{5}} \sin {\left (3 t \right )} \]