70.28.7 problem 8

Internal problem ID [19160]
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 : 8
Date solved : Thursday, October 02, 2025 at 03:38:49 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.385 (sec). Leaf size: 75

ode:=[diff(x__1(t),t) = -4*x__1(t)+x__2(t)+3*x__3(t)+3*t, diff(x__2(t),t) = -2*x__2(t), diff(x__3(t),t) = -2*x__1(t)+x__2(t)+x__3(t)+3*cos(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -{\mathrm e}^{-2 t} c_1 +\frac {15}{4}-\frac {3 t}{2}+\frac {9 \cos \left (t \right )}{10}+\frac {27 \sin \left (t \right )}{10}+{\mathrm e}^{-t} c_2 \\ x_{2} \left (t \right ) &= c_3 \,{\mathrm e}^{-2 t} \\ x_{3} \left (t \right ) &= -\frac {2 \,{\mathrm e}^{-2 t} c_1}{3}+\frac {9}{2}+\frac {33 \sin \left (t \right )}{10}+\frac {21 \cos \left (t \right )}{10}+{\mathrm e}^{-t} c_2 -3 t -\frac {c_3 \,{\mathrm e}^{-2 t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.138 (sec). Leaf size: 271

ode={D[x1[t],t]==-4*x1[t]+1*x2[t]+3*x3[t]+3*t,D[x2[t],t]==0*x1[t]-2*x2[t]-0*x3[t]+0,D[x3[t],t]==-2*x1[t]+1*x2[t]+1*x3[t]+3*Cos[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to e^{-2 t} \left (\left (3-2 e^t\right ) \int _1^t3 e^{K[1]} \left (\left (-2+3 e^{K[1]}\right ) K[1]-3 \left (-1+e^{K[1]}\right ) \cos (K[1])\right )dK[1]+3 \left (e^t-1\right ) \int _1^t3 e^{K[2]} \left (\left (3-2 e^{K[2]}\right ) \cos (K[2])+2 \left (-1+e^{K[2]}\right ) K[2]\right )dK[2]-2 c_1 e^t+c_2 e^t+3 c_3 e^t+3 c_1-c_2-3 c_3\right )\\ \text {x2}(t)&\to c_2 e^{-2 t}\\ \text {x3}(t)&\to e^{-2 t} \left (-2 \left (e^t-1\right ) \int _1^t3 e^{K[1]} \left (\left (-2+3 e^{K[1]}\right ) K[1]-3 \left (-1+e^{K[1]}\right ) \cos (K[1])\right )dK[1]+\left (3 e^t-2\right ) \int _1^t3 e^{K[2]} \left (\left (3-2 e^{K[2]}\right ) \cos (K[2])+2 \left (-1+e^{K[2]}\right ) K[2]\right )dK[2]-2 c_1 e^t+c_2 e^t+3 c_3 e^t+2 c_1-c_2-2 c_3\right ) \end{align*}

✓ Sympy. Time used: 0.179 (sec). Leaf size: 85

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-3*t + 4*x__1(t) - x__2(t) - 3*x__3(t) + Derivative(x__1(t), t),0),Eq(2*x__2(t) + Derivative(x__2(t), t),0),Eq(2*x__1(t) - x__2(t) - x__3(t) - 3*cos(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{- t} - \frac {3 t}{2} + \left (\frac {C_{2}}{2} + \frac {3 C_{3}}{2}\right ) e^{- 2 t} + \frac {27 \sin {\left (t \right )}}{10} + \frac {9 \cos {\left (t \right )}}{10} + \frac {15}{4}, \ x^{2}{\left (t \right )} = C_{2} e^{- 2 t}, \ x^{3}{\left (t \right )} = C_{1} e^{- t} + C_{3} e^{- 2 t} - 3 t + \frac {33 \sin {\left (t \right )}}{10} + \frac {21 \cos {\left (t \right )}}{10} + \frac {9}{2}\right ] \]