8.25.2 problem Example 2, page 364
Internal
problem
ID
[2759]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Section
3.12,
Systems
of
differential
equations.
The
nonhomogeneous
equation.
variation
of
parameters.
Page
366
Problem
number
:
Example
2,
page
364
Date
solved
:
Tuesday, September 30, 2025 at 05:50:41 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+{\mathrm e}^{c t}\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.644 (sec). Leaf size: 407
ode:=[diff(x__1(t),t) = x__1(t)+exp(c*t), diff(x__2(t),t) = 2*x__1(t)+x__2(t)-2*x__3(t), diff(x__3(t),t) = 3*x__1(t)+2*x__2(t)+x__3(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= {\mathrm e}^{t} c_3 +\frac {{\mathrm e}^{c t}}{c -1} \\
x_{2} \left (t \right ) &= \frac {2 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_2 \,c^{3}+2 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_1 \,c^{3}-3 c^{3} {\mathrm e}^{t} c_3 \cos \left (2 t \right )-6 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_2 \,c^{2}-6 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_1 \,c^{2}+9 c^{2} {\mathrm e}^{t} c_3 \cos \left (2 t \right )-3 c^{3} {\mathrm e}^{t} c_3 +14 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_2 c +14 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_1 c -21 \,{\mathrm e}^{t} c_3 c \cos \left (2 t \right )+9 c^{2} {\mathrm e}^{t} c_3 -10 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_2 -10 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_1 +15 \,{\mathrm e}^{t} c_3 \cos \left (2 t \right )-21 \,{\mathrm e}^{t} c_3 c +4 c \,{\mathrm e}^{t +t \left (c -1\right )}+15 \,{\mathrm e}^{t} c_3 -16 \,{\mathrm e}^{t +t \left (c -1\right )}}{2 \left (c -1\right ) \left (c^{2}-2 c +5\right )} \\
x_{3} \left (t \right ) &= -\frac {2 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_2 \,c^{3}-2 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_1 \,c^{3}+3 \,{\mathrm e}^{t} c_3 \,c^{3} \sin \left (2 t \right )-6 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_2 \,c^{2}+6 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_1 \,c^{2}-9 \,{\mathrm e}^{t} c_3 \,c^{2} \sin \left (2 t \right )-2 c^{3} {\mathrm e}^{t} c_3 +14 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_2 c -14 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_1 c +21 \,{\mathrm e}^{t} c_3 c \sin \left (2 t \right )+6 c^{2} {\mathrm e}^{t} c_3 -10 \,{\mathrm e}^{t} \cos \left (2 t \right ) c_2 +10 \,{\mathrm e}^{t} \sin \left (2 t \right ) c_1 -15 c_3 \,{\mathrm e}^{t} \sin \left (2 t \right )-14 \,{\mathrm e}^{t} c_3 c +10 \,{\mathrm e}^{t} c_3 -6 \,{\mathrm e}^{c t} c -2 \,{\mathrm e}^{c t}}{2 \left (c -1\right ) \left (c^{2}-2 c +5\right )} \\
\end{align*}
✓ Mathematica. Time used: 0.259 (sec). Leaf size: 256
ode={D[ x1[t],t]==1*x1[t]+0*x2[t]+0*x3[t]+Exp[c*t],D[ x2[t],t]==2*x1[t]+1*x2[t]-2*x3[t],D[ x3[t],t]==3*x1[t]+2*x2[t]+1*x3[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {x1}(t)&\to e^t \left (\frac {e^{(c-1) t}}{c-1}+c_1\right )\\ \text {x2}(t)&\to \frac {e^t \left (-3 c^3 c_1+9 c^2 c_1+\left (c^3-3 c^2+7 c-5\right ) (3 c_1+2 c_2) \cos (2 t)+2 \left (c^3-3 c^2+7 c-5\right ) (c_1-c_3) \sin (2 t)+4 c e^{(c-1) t}-16 e^{(c-1) t}-21 c c_1+15 c_1\right )}{2 (c-1) \left (c^2-2 c+5\right )}\\ \text {x3}(t)&\to \frac {e^t \left (-2 \left (c^3-3 c^2+7 c-5\right ) (c_1-c_3) \cos (2 t)+\left (c^3-3 c^2+7 c-5\right ) (3 c_1+2 c_2) \sin (2 t)+2 \left (c^3-3 c^2+7 c-5\right ) c_1+2 (3 c+1) e^{(c-1) t}\right )}{2 (c-1) \left (c^2-2 c+5\right )} \end{align*}
✓ Sympy. Time used: 12.128 (sec). Leaf size: 860
from sympy import *
t = symbols("t")
c = symbols("c")
x__1 = Function("x__1")
x__2 = Function("x__2")
x__3 = Function("x__3")
ode=[Eq(-x__1(t) - exp(c*t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - x__2(t) + 2*x__3(t) + Derivative(x__2(t), t),0),Eq(-3*x__1(t) - 2*x__2(t) - x__3(t) + Derivative(x__3(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
\[
\text {Solution too large to show}
\]