70.28.8 problem 9

Internal problem ID [19161]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.6 (Nonhomogeneous Linear Systems). Problems at page 436
Problem number : 9
Date solved : Thursday, October 02, 2025 at 03:38:50 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.293 (sec). Leaf size: 81

ode:=[diff(x__1(t),t) = -1/2*x__1(t)+x__2(t)+1/2*x__3(t), diff(x__2(t),t) = x__1(t)-x__2(t)+x__3(t)-sin(t), diff(x__3(t),t) = 1/2*x__1(t)+x__2(t)-1/2*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {\cos \left (t \right )}{10}+\frac {3 \sin \left (t \right )}{10}+c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-2 t}+c_3 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= -\frac {\sin \left (t \right )}{10}+\frac {3 \cos \left (t \right )}{10}+c_1 \,{\mathrm e}^{t}-2 c_2 \,{\mathrm e}^{-2 t} \\ x_{3} \left (t \right ) &= \frac {\cos \left (t \right )}{10}+\frac {3 \sin \left (t \right )}{10}+c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-2 t}-c_3 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.126 (sec). Leaf size: 549

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

\begin{align*} \text {x1}(t)&\to \frac {1}{6} e^{-2 t} \left (\left (3 e^t+2 e^{3 t}+1\right ) \int _1^t\frac {1}{3} e^{-K[1]} \left (-1+e^{3 K[1]}\right ) \sin (K[1])dK[1]+2 \left (e^{3 t}-1\right ) \int _1^t-\frac {1}{3} e^{2 K[2]} \left (2+e^{-3 K[2]}\right ) \sin (K[2])dK[2]+\left (-3 e^t+2 e^{3 t}+1\right ) \int _1^t\frac {1}{3} e^{-K[3]} \left (-1+e^{3 K[3]}\right ) \sin (K[3])dK[3]+c_1 \left (3 e^t+2 e^{3 t}+1\right )+2 c_2 \left (e^{3 t}-1\right )+c_3 \left (-3 e^t+2 e^{3 t}+1\right )\right )\\ \text {x2}(t)&\to \frac {1}{3} e^{-2 t} \left (\left (e^{3 t}-1\right ) \int _1^t\frac {1}{3} e^{-K[1]} \left (-1+e^{3 K[1]}\right ) \sin (K[1])dK[1]+\left (e^{3 t}+2\right ) \int _1^t-\frac {1}{3} e^{2 K[2]} \left (2+e^{-3 K[2]}\right ) \sin (K[2])dK[2]+e^{3 t} \int _1^t\frac {1}{3} e^{-K[3]} \left (-1+e^{3 K[3]}\right ) \sin (K[3])dK[3]-\int _1^t\frac {1}{3} e^{-K[3]} \left (-1+e^{3 K[3]}\right ) \sin (K[3])dK[3]+c_1 e^{3 t}+c_2 e^{3 t}+c_3 e^{3 t}-c_1+2 c_2-c_3\right )\\ \text {x3}(t)&\to \frac {1}{6} e^{-2 t} \left (\left (-3 e^t+2 e^{3 t}+1\right ) \int _1^t\frac {1}{3} e^{-K[1]} \left (-1+e^{3 K[1]}\right ) \sin (K[1])dK[1]+2 \left (e^{3 t}-1\right ) \int _1^t-\frac {1}{3} e^{2 K[2]} \left (2+e^{-3 K[2]}\right ) \sin (K[2])dK[2]+\left (3 e^t+2 e^{3 t}+1\right ) \int _1^t\frac {1}{3} e^{-K[3]} \left (-1+e^{3 K[3]}\right ) \sin (K[3])dK[3]+c_1 \left (-3 e^t+2 e^{3 t}+1\right )+2 c_2 \left (e^{3 t}-1\right )+c_3 \left (3 e^t+2 e^{3 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.214 (sec). Leaf size: 88

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)/2 - x__2(t) - x__3(t)/2 + Derivative(x__1(t), t),0),Eq(-x__1(t) + x__2(t) - x__3(t) + sin(t) + Derivative(x__2(t), t),0),Eq(-x__1(t)/2 - x__2(t) + x__3(t)/2 + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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