12.21.13 problem section 10.4, problem 13
Internal
problem
ID
[2251]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
10
Linear
system
of
Differential
equations.
Section
10.4,
constant
coefficient
homogeneous
system.
Page
540
Problem
number
:
section
10.4,
problem
13
Date
solved
:
Monday, January 27, 2025 at 05:44:07 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-2 y_{1} \left (t \right )+2 y_{2} \left (t \right )-6 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=2 y_{1} \left (t \right )+6 y_{2} \left (t \right )+2 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-2 y_{1} \left (t \right )-2 y_{2} \left (t \right )+2 y_{3} \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.040 (sec). Leaf size: 65
dsolve([diff(y__1(t),t)=-2*y__1(t)+2*y__2(t)-6*y__3(t),diff(y__2(t),t)=2*y__1(t)+6*y__2(t)+2*y__3(t),diff(y__3(t),t)=-2*y__1(t)-2*y__2(t)+2*y__3(t)],singsol=all)
\begin{align*}
y_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{6 t}+c_2 \,{\mathrm e}^{-4 t}+c_3 \,{\mathrm e}^{4 t} \\
y_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{6 t}-\frac {c_2 \,{\mathrm e}^{-4 t}}{4} \\
y_{3} \left (t \right ) &= -c_1 \,{\mathrm e}^{6 t}+\frac {c_2 \,{\mathrm e}^{-4 t}}{4}-c_3 \,{\mathrm e}^{4 t} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 257
DSolve[{D[ y1[t],t]==-2*y1[t]+2*y2[t]-6*y3[t],D[ y2[t],t]==2*y1[t]+6*y2[t]+2*y3[t],D[ y1[t],t]==-2*y1[t]-2*y2[t]+2*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {y1}(t)\to -\frac {e^{-\frac {1}{2} \left (\sqrt {73}-5\right ) t} \left (2 c_1 \left (\left (841 \sqrt {73}-7227\right ) e^{\sqrt {73} t}-7227-841 \sqrt {73}\right )+c_2 \left (\left (171 \sqrt {73}-1825\right ) e^{\sqrt {73} t}-1825-171 \sqrt {73}\right )\right )}{598016} \\
\text {y2}(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {73}-5\right ) t} \left (c_1 \left (\left (342 \sqrt {73}-3650\right ) e^{\sqrt {73} t}-3650-342 \sqrt {73}\right )-c_2 \left (\left (1971+143 \sqrt {73}\right ) e^{\sqrt {73} t}+1971-143 \sqrt {73}\right )\right )}{598016} \\
\text {y3}(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {73}-5\right ) t} \left (c_1 \left (\left (342 \sqrt {73}-3650\right ) e^{\sqrt {73} t}-3650-342 \sqrt {73}\right )-c_2 \left (\left (1971+143 \sqrt {73}\right ) e^{\sqrt {73} t}+1971-143 \sqrt {73}\right )\right )}{1196032} \\
\end{align*}