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
:
Monday, January 27, 2025 at 09:25:42 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x_{1}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right )-x_{3} \left (t \right )+{\mathrm e}^{3 t}\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )-x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.585 (sec). Leaf size: 163
dsolve([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)],singsol=all)
\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*}
✓ Solution by Mathematica
Time used: 0.135 (sec). Leaf size: 187
DSolve[{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]},{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*}