14.26.14 problem 14

Internal problem ID [2787]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 3. Systems of differential equations. Section 3.13 (Solving systems by Laplace transform). Page 370
Problem number : 14
Date solved : Tuesday, March 04, 2025 at 02:42:38 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right )+{\mathrm e}^{t} \cos \left (2 t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 1\\ x_{3} \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.225 (sec). Leaf size: 59

ode:=[diff(x__1(t),t) = x__1(t), diff(x__2(t),t) = 2*x__1(t)+x__2(t)-2*x__3(t), diff(x__3(t),t) = 3*x__1(t)+2*x__2(t)+x__3(t)+exp(t)*cos(2*t)]; 
ic:=x__1(0) = 0x__2(0) = 1x__3(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= 0 \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-2 \sin \left (2 t \right )-t \sin \left (2 t \right )+2 \cos \left (2 t \right )\right )}{2} \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (5 \sin \left (2 t \right )+4 \cos \left (2 t \right )+2 t \cos \left (2 t \right )\right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 58

ode={D[x1[t],t]==1*x1[t]+0*x2[t]+0*x3[t],D[x2[t],t]==2*x1[t]+1*x2[t]-2*x3[t],D[x3[t],t]==3*x1[t]+2*x2[t]+1*x3[t]+Exp[t]*Cos[2*t]}; 
ic={x1[0]==0,x2[0]==1,x3[0]==1}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to 0 \\ \text {x2}(t)\to \frac {1}{2} e^t (2 \cos (2 t)-(t+2) \sin (2 t)) \\ \text {x3}(t)\to \frac {1}{4} e^t (5 \sin (2 t)+2 (t+2) \cos (2 t)) \\ \end{align*}

✓ Sympy. Time used: 0.221 (sec). Leaf size: 119

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-x__1(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - x__2(t) + 2*x__3(t) + Derivative(x__2(t), t),0),Eq(-3*x__1(t) - 2*x__2(t) - x__3(t) - exp(t)*cos(2*t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{t}, \ x^{2}{\left (t \right )} = - \frac {3 C_{1} e^{t}}{2} - C_{2} e^{t} \sin {\left (2 t \right )} - C_{3} e^{t} \cos {\left (2 t \right )} - \frac {t e^{t} \sin {\left (2 t \right )}}{2}, \ x^{3}{\left (t \right )} = C_{1} e^{t} + C_{2} e^{t} \cos {\left (2 t \right )} - C_{3} e^{t} \sin {\left (2 t \right )} + \frac {t e^{t} \cos {\left (2 t \right )}}{2} + \frac {e^{t} \sin ^{3}{\left (2 t \right )}}{4} + \frac {e^{t} \sin {\left (2 t \right )} \cos ^{2}{\left (2 t \right )}}{4}\right ] \]