76.28.5 problem 6

Internal problem ID [17793]
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, March 13, 2025 at 10:49:23 AM
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: 0.182 (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 ) &= c_{1} {\mathrm e}^{2 t}+\frac {{\mathrm e}^{-t}}{6}-\frac {9}{4}-\frac {t}{2}+c_{2} {\mathrm e}^{t} \\ x_{2} \left (t \right ) &= -\frac {t}{2}-\frac {1}{4}+c_{3} {\mathrm e}^{2 t} \\ x_{3} \left (t \right ) &= 2 c_{1} {\mathrm e}^{2 t}-\frac {{\mathrm e}^{-t}}{6}-\frac {7}{4}+c_{2} {\mathrm e}^{t}-\frac {t}{2}+c_{3} {\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.446 (sec). Leaf size: 130

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 \frac {1}{4} (-2 t-9)+\frac {e^{-t}}{6}-(c_1+c_2-c_3) e^{2 t}+(2 c_1+c_2-c_3) e^t \\ \text {x2}(t)\to -\frac {t}{2}+c_2 e^{2 t}-\frac {1}{4} \\ \text {x3}(t)\to \frac {1}{4} (-2 t-7)-\frac {e^{-t}}{6}-(2 c_1+c_2-2 c_3) e^{2 t}+(2 c_1+c_2-c_3) e^t \\ \end{align*}

✓ Sympy. Time used: 0.278 (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)
 

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