28.4.39 problem 7.39

Internal problem ID [4571]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.39
Date solved : Sunday, March 30, 2025 at 03:26:32 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )+26 \sin \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{1} \left (t \right )+4 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.217 (sec). Leaf size: 47

ode:=[diff(x__1(t),t) = 2*x__1(t)+x__2(t)+26*sin(t), diff(x__2(t),t) = 3*x__1(t)+4*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{5 t} c_2}{3}-{\mathrm e}^{t} c_1 -11 \sin \left (t \right )-10 \cos \left (t \right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{5 t} c_2 +{\mathrm e}^{t} c_1 +9 \cos \left (t \right )+6 \sin \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.116 (sec). Leaf size: 85

ode={D[x1[t],t]==2*x1[t]+x2[t]+26*Sin[t],D[x2[t],t]==3*x1[t]+4*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to -11 \sin (t)-10 \cos (t)+\frac {1}{4} e^t \left (c_1 \left (e^{4 t}+3\right )+c_2 \left (e^{4 t}-1\right )\right ) \\ \text {x2}(t)\to 6 \sin (t)+9 \cos (t)+\frac {1}{4} e^t \left (3 c_1 \left (e^{4 t}-1\right )+c_2 \left (3 e^{4 t}+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.323 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-2*x__1(t) - x__2(t) - 26*sin(t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) - 4*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{1} e^{t} + \frac {C_{2} e^{5 t}}{3} - 11 \sin {\left (t \right )} - 10 \cos {\left (t \right )}, \ x^{2}{\left (t \right )} = C_{1} e^{t} + C_{2} e^{5 t} + 6 \sin {\left (t \right )} + 9 \cos {\left (t \right )}\right ] \]