70.28.6 problem 7
Internal
problem
ID
[19159]
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
:
7
Date
solved
:
Thursday, October 02, 2025 at 03:38:48 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\frac {x_{1} \left (t \right )}{2}+\frac {x_{2} \left (t \right )}{2}-\frac {x_{3} \left (t \right )}{2}+1\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )-2 x_{2} \left (t \right )+x_{3} \left (t \right )+t\\ \frac {d}{d t}x_{3} \left (t \right )&=\frac {x_{1} \left (t \right )}{2}+\frac {x_{2} \left (t \right )}{2}-\frac {3 x_{3} \left (t \right )}{2}+11 \,{\mathrm e}^{-3 t} \end{align*}
✓ Maple. Time used: 0.186 (sec). Leaf size: 96
ode:=[diff(x__1(t),t) = -1/2*x__1(t)+1/2*x__2(t)-1/2*x__3(t)+1, diff(x__2(t),t) = -x__1(t)-2*x__2(t)+x__3(t)+t, diff(x__3(t),t) = 1/2*x__1(t)+1/2*x__2(t)-3/2*x__3(t)+11*exp(-3*t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= -{\mathrm e}^{-2 t} c_2 -\frac {11 \,{\mathrm e}^{-3 t}}{4}+\frac {t}{4}+\frac {7}{8}+{\mathrm e}^{-t} c_3 \\
x_{2} \left (t \right ) &= 2 \,{\mathrm e}^{-2 t} c_2 +\frac {11 \,{\mathrm e}^{-3 t}}{2}+\frac {t}{2}-\frac {3}{4}-2 \,{\mathrm e}^{-t} c_3 +{\mathrm e}^{-t} c_1 \\
x_{3} \left (t \right ) &= -{\mathrm e}^{-2 t} c_2 -\frac {33 \,{\mathrm e}^{-3 t}}{4}-\frac {1}{8}-{\mathrm e}^{-t} c_3 +\frac {t}{4}+{\mathrm e}^{-t} c_1 \\
\end{align*}
✓ Mathematica. Time used: 0.804 (sec). Leaf size: 720
ode={D[x1[t],t]==-1/2*x1[t]+1/2*x2[t]-1/2*x3[t]+1,D[x2[t],t]==-1*x1[t]-2*x2[t]+1*x3[t]+t,D[x3[t],t]==1/2*x1[t]+1/2*x2[t]-3/2*x3[t]+11*Exp[-3*t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {x1}(t)&\to \frac {1}{2} e^{-2 t} \left (\left (3 e^t-1\right ) \int _1^t\frac {1}{2} e^{-2 K[1]} \left (-e^{4 K[1]} (K[1]+1)+11 e^{K[1]}+e^{3 K[1]} (K[1]+3)-11\right )dK[1]+\left (e^t-1\right ) \int _1^te^{-2 K[2]} \left (e^{4 K[2]} (K[2]+1)-11 e^{K[2]}-e^{3 K[2]}+11\right )dK[2]-e^t \int _1^t\frac {1}{2} e^{-2 K[3]} \left (e^{3 K[3]} (K[3]+1)-e^{4 K[3]} (K[3]+1)+11 e^{K[3]}+11\right )dK[3]+\int _1^t\frac {1}{2} e^{-2 K[3]} \left (e^{3 K[3]} (K[3]+1)-e^{4 K[3]} (K[3]+1)+11 e^{K[3]}+11\right )dK[3]+3 c_1 e^t+c_2 e^t-c_3 e^t-c_1-c_2+c_3\right )\\ \text {x2}(t)&\to e^{-2 t} \left (-\left (e^t-1\right ) \int _1^t\frac {1}{2} e^{-2 K[1]} \left (-e^{4 K[1]} (K[1]+1)+11 e^{K[1]}+e^{3 K[1]} (K[1]+3)-11\right )dK[1]+\int _1^te^{-2 K[2]} \left (e^{4 K[2]} (K[2]+1)-11 e^{K[2]}-e^{3 K[2]}+11\right )dK[2]+e^t \int _1^t\frac {1}{2} e^{-2 K[3]} \left (e^{3 K[3]} (K[3]+1)-e^{4 K[3]} (K[3]+1)+11 e^{K[3]}+11\right )dK[3]-\int _1^t\frac {1}{2} e^{-2 K[3]} \left (e^{3 K[3]} (K[3]+1)-e^{4 K[3]} (K[3]+1)+11 e^{K[3]}+11\right )dK[3]+c_1 \left (-e^t\right )+c_3 e^t+c_1+c_2-c_3\right )\\ \text {x3}(t)&\to \frac {1}{2} e^{-2 t} \left (\left (e^t-1\right ) \int _1^t\frac {1}{2} e^{-2 K[1]} \left (-e^{4 K[1]} (K[1]+1)+11 e^{K[1]}+e^{3 K[1]} (K[1]+3)-11\right )dK[1]+\left (e^t-1\right ) \int _1^te^{-2 K[2]} \left (e^{4 K[2]} (K[2]+1)-11 e^{K[2]}-e^{3 K[2]}+11\right )dK[2]+e^t \int _1^t\frac {1}{2} e^{-2 K[3]} \left (e^{3 K[3]} (K[3]+1)-e^{4 K[3]} (K[3]+1)+11 e^{K[3]}+11\right )dK[3]+\int _1^t\frac {1}{2} e^{-2 K[3]} \left (e^{3 K[3]} (K[3]+1)-e^{4 K[3]} (K[3]+1)+11 e^{K[3]}+11\right )dK[3]+c_1 e^t+c_2 e^t+c_3 e^t-c_1-c_2+c_3\right ) \end{align*}
✓ Sympy. Time used: 0.216 (sec). Leaf size: 90
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)/2 + x__3(t)/2 + Derivative(x__1(t), t) - 1,0),Eq(-t + x__1(t) + 2*x__2(t) - x__3(t) + Derivative(x__2(t), t),0),Eq(-x__1(t)/2 - x__2(t)/2 + 3*x__3(t)/2 + Derivative(x__3(t), t) - 11*exp(-3*t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = C_{3} e^{- 2 t} + \frac {t}{4} - \left (C_{1} - C_{2}\right ) e^{- t} + \frac {7}{8} - \frac {11 e^{- 3 t}}{4}, \ x^{2}{\left (t \right )} = C_{1} e^{- t} - 2 C_{3} e^{- 2 t} + \frac {t}{2} - \frac {3}{4} + \frac {11 e^{- 3 t}}{2}, \ x^{3}{\left (t \right )} = C_{2} e^{- t} + C_{3} e^{- 2 t} + \frac {t}{4} - \frac {1}{8} - \frac {33 e^{- 3 t}}{4}\right ]
\]