9.4.32 problem problem 43
Internal
problem
ID
[996]
Book
:
Differential
equations
and
linear
algebra,
4th
ed.,
Edwards
and
Penney
Section
:
Section
7.3,
The
eigenvalue
method
for
linear
systems.
Page
395
Problem
number
:
problem
43
Date
solved
:
Monday, January 27, 2025 at 03:22:47 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-20 x_{1} \left (t \right )+11 x_{2} \left (t \right )+13 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=12 x_{1} \left (t \right )-x_{2} \left (t \right )-7 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-48 x_{1} \left (t \right )+21 x_{2} \left (t \right )+31 x_{3} \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.029 (sec). Leaf size: 71
dsolve([diff(x__1(t),t)=-20*x__1(t)+11*x__2(t)+13*x__3(t),diff(x__2(t),t)=12*x__1(t)-1*x__2(t)-7*x__3(t),diff(x__3(t),t)=-48*x__1(t)+21*x__2(t)+31*x__3(t)],singsol=all)
\begin{align*}
x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-2 t}+c_2 \,{\mathrm e}^{8 t}+c_3 \,{\mathrm e}^{4 t} \\
x_{2} \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{-2 t}}{3}-c_2 \,{\mathrm e}^{8 t}+c_3 \,{\mathrm e}^{4 t} \\
x_{3} \left (t \right ) &= \frac {5 c_1 \,{\mathrm e}^{-2 t}}{3}+3 c_2 \,{\mathrm e}^{8 t}+c_3 \,{\mathrm e}^{4 t} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 554
DSolve[{D[ x1[t],t]==20*x1[t]+11*x2[t]+13*x3[t],D[ x2[t],t]==12*x1[t]-1*x2[t]-7*x3[t],D[ x3[t],t]==-48*x1[t]+21*x2[t]+31*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {x1}(t)\to c_2 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {11 \text {$\#$1} e^{\text {$\#$1} t}-68 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {13 \text {$\#$1} e^{\text {$\#$1} t}-64 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-30 \text {$\#$1} e^{\text {$\#$1} t}+116 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ] \\
\text {x2}(t)\to 12 c_1 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]-c_3 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {7 \text {$\#$1} e^{\text {$\#$1} t}-296 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-51 \text {$\#$1} e^{\text {$\#$1} t}+1244 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ] \\
\text {x3}(t)\to -12 c_1 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {4 \text {$\#$1} e^{\text {$\#$1} t}-17 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]+3 c_2 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {7 \text {$\#$1} e^{\text {$\#$1} t}-316 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-19 \text {$\#$1} e^{\text {$\#$1} t}-152 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ] \\
\end{align*}