28.4.53 problem 7.53
Internal
problem
ID
[4585]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
7.
Systems
of
linear
differential
equations.
Problems
at
page
351
Problem
number
:
7.53
Date
solved
:
Tuesday, March 04, 2025 at 06:56:01 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 )-x_{3} \left (t \right )+{\mathrm e}^{3 t}\\ \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 )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.585 (sec). Leaf size: 163
ode:=[diff(x__1(t),t) = 4*x__1(t)-x__2(t)-x__3(t)+exp(3*t), diff(x__2(t),t) = x__1(t)+2*x__2(t)-x__3(t), diff(x__3(t),t) = x__1(t)+x__2(t)+2*x__3(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= {\mathrm e}^{\frac {5 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_3 +{\mathrm e}^{\frac {5 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2} +\frac {{\mathrm e}^{3 t}}{2}+{\mathrm e}^{3 t} t +c_{1} {\mathrm e}^{3 t} \\
x_{2} \left (t \right ) &= {\mathrm e}^{\frac {5 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_3 +{\mathrm e}^{\frac {5 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2} +\frac {{\mathrm e}^{3 t}}{2} \\
x_{3} \left (t \right ) &= \frac {{\mathrm e}^{\frac {5 t}{2}} \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{2}-\frac {{\mathrm e}^{\frac {5 t}{2}} \sqrt {7}\, \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_3}{2}+\frac {{\mathrm e}^{\frac {5 t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_3}{2}+\frac {{\mathrm e}^{\frac {5 t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2}}{2}+c_{1} {\mathrm e}^{3 t}+{\mathrm e}^{3 t} t \\
\end{align*}
✓ Mathematica. Time used: 0.135 (sec). Leaf size: 187
ode={D[x1[t],t]==4*x1[t]-x2[t]-x3[t]+Exp[3*t],D[x2[t],t]==x1[t]+2*x2[t]-x3[t],D[x3[t],t]==x1[t]+x2[t]+3*x3[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to \frac {1}{2} e^{3 t} \left (t+(c_1+c_2) \cos \left (\sqrt {2} t\right )+\sqrt {2} (c_1-c_2-c_3) \sin \left (\sqrt {2} t\right )+1+c_1-c_2\right ) \\
\text {x2}(t)\to \frac {1}{2} e^{3 t} \left (-t+(c_1+c_2) \cos \left (\sqrt {2} t\right )+\sqrt {2} (c_1-c_2-c_3) \sin \left (\sqrt {2} t\right )+1-c_1+c_2\right ) \\
\text {x3}(t)\to \frac {1}{2} e^{3 t} \left (2 t+2 (-c_1+c_2+c_3) \cos \left (\sqrt {2} t\right )+\sqrt {2} (c_1+c_2) \sin \left (\sqrt {2} t\right )+1+2 c_1-2 c_2\right ) \\
\end{align*}
✓ Sympy. Time used: 0.547 (sec). Leaf size: 260
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
x__3 = Function("x__3")
ode=[Eq(-4*x__1(t) + x__2(t) + x__3(t) - exp(3*t) + Derivative(x__1(t), t),0),Eq(-x__1(t) - 2*x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(-x__1(t) - x__2(t) - 2*x__3(t) + Derivative(x__3(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = C_{1} e^{3 t} + t e^{3 t} - \left (\frac {C_{2}}{4} + \frac {\sqrt {7} C_{3}}{4}\right ) e^{\frac {5 t}{2}} \sin {\left (\frac {\sqrt {7} t}{2} \right )} - \left (\frac {\sqrt {7} C_{2}}{4} - \frac {C_{3}}{4}\right ) e^{\frac {5 t}{2}} \cos {\left (\frac {\sqrt {7} t}{2} \right )} + \frac {e^{3 t} \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2} + \frac {e^{3 t} \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2}, \ x^{2}{\left (t \right )} = - \left (\frac {C_{2}}{4} + \frac {\sqrt {7} C_{3}}{4}\right ) e^{\frac {5 t}{2}} \sin {\left (\frac {\sqrt {7} t}{2} \right )} - \left (\frac {\sqrt {7} C_{2}}{4} - \frac {C_{3}}{4}\right ) e^{\frac {5 t}{2}} \cos {\left (\frac {\sqrt {7} t}{2} \right )} + \frac {e^{3 t} \sin ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2} + \frac {e^{3 t} \cos ^{2}{\left (\frac {\sqrt {7} t}{2} \right )}}{2}, \ x^{3}{\left (t \right )} = C_{1} e^{3 t} - C_{2} e^{\frac {5 t}{2}} \sin {\left (\frac {\sqrt {7} t}{2} \right )} + C_{3} e^{\frac {5 t}{2}} \cos {\left (\frac {\sqrt {7} t}{2} \right )} + t e^{3 t}\right ]
\]