14.25.1 problem Example 1, page 361

Internal problem ID [2758]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.12, Systems of differential equations. The nonhomogeneous equation. variation of parameters. Page 366
Problem number : Example 1, page 361
Date solved : Tuesday, March 04, 2025 at 02:40:58 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*}

✓ Maple. Time used: 1.038 (sec). Leaf size: 90

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)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_3 \,{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-3 c_3 -3 c_3 \cos \left (2 t \right )+2 c_2 \sin \left (2 t \right )-t \sin \left (2 t \right )+2 c_1 \cos \left (2 t \right )\right )}{2} \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (4 c_1 \sin \left (2 t \right )-6 c_3 \sin \left (2 t \right )-4 c_2 \cos \left (2 t \right )+2 t \cos \left (2 t \right )+\sin \left (2 t \right )+4 c_3 \right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 103

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={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 e^t \\ \text {x2}(t)\to -\frac {1}{8} e^t ((1-12 c_1-8 c_2) \cos (2 t)+4 (t-2 c_1+2 c_3) \sin (2 t)+12 c_1) \\ \text {x3}(t)\to \frac {1}{8} e^t (4 (t-2 c_1+2 c_3) \cos (2 t)+(1+12 c_1+8 c_2) \sin (2 t)+8 c_1) \\ \end{align*}

✓ Sympy. Time used: 0.174 (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)
 

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