70.28.5 problem 6

Internal problem ID [19158]
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 : 6
Date solved : Thursday, October 02, 2025 at 03:38:47 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.193 (sec). Leaf size: 70

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} c_1 +\frac {{\mathrm e}^{-t}}{6}-\frac {9}{4}-\frac {t}{2}+{\mathrm e}^{t} c_2 \\ x_{2} \left (t \right ) &= -\frac {t}{2}-\frac {1}{4}+{\mathrm e}^{2 t} c_3 \\ x_{3} \left (t \right ) &= 2 \,{\mathrm e}^{2 t} c_1 -\frac {{\mathrm e}^{-t}}{6}-\frac {7}{4}+{\mathrm e}^{t} c_2 -\frac {t}{2}+{\mathrm e}^{2 t} c_3 \\ \end{align*}

✓ Mathematica. Time used: 0.347 (sec). Leaf size: 322

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

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

✓ Sympy. Time used: 0.167 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(x__2(t) - x__3(t) + Derivative(x__1(t), t) - 1,0),Eq(-t - 2*x__2(t) + Derivative(x__2(t), t),0),Eq(2*x__1(t) + x__2(t) - 3*x__3(t) + Derivative(x__3(t), t) - exp(-t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{t} - \frac {t}{2} - \left (\frac {C_{2}}{2} - \frac {C_{3}}{2}\right ) e^{2 t} - \frac {9}{4} + \frac {e^{- t}}{6}, \ x^{2}{\left (t \right )} = C_{2} e^{2 t} - \frac {t}{2} - \frac {1}{4}, \ x^{3}{\left (t \right )} = C_{1} e^{t} + C_{3} e^{2 t} - \frac {t}{2} - \frac {7}{4} - \frac {e^{- t}}{6}\right ] \]