9.6.34 problem problem 34
Internal
problem
ID
[1041]
Book
:
Differential
equations
and
linear
algebra,
4th
ed.,
Edwards
and
Penney
Section
:
Section
7.6,
Multiple
Eigenvalue
Solutions.
Page
451
Problem
number
:
problem
34
Date
solved
:
Saturday, March 29, 2025 at 10:37:32 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-8 x_{3} \left (t \right )-3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-18 x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-9 x_{1} \left (t \right )-3 x_{2} \left (t \right )-25 x_{3} \left (t \right )-9 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=33 x_{1} \left (t \right )+10 x_{2} \left (t \right )+90 x_{3} \left (t \right )+32 x_{4} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.162 (sec). Leaf size: 251
ode:=[diff(x__1(t),t) = 2*x__1(t)-8*x__3(t)-3*x__4(t), diff(x__2(t),t) = -18*x__1(t)-x__2(t), diff(x__3(t),t) = -9*x__1(t)-3*x__2(t)-25*x__3(t)-9*x__4(t), diff(x__4(t),t) = 33*x__1(t)+10*x__2(t)+90*x__3(t)+32*x__4(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (3 \sin \left (3 t \right ) c_3 t -3 \sin \left (3 t \right ) c_4 t +3 \cos \left (3 t \right ) c_3 t +3 \cos \left (3 t \right ) c_4 t +3 \sin \left (3 t \right ) c_1 -3 \sin \left (3 t \right ) c_2 +\sin \left (3 t \right ) c_3 +3 \cos \left (3 t \right ) c_1 +3 \cos \left (3 t \right ) c_2 +\cos \left (3 t \right ) c_4 \right )}{18} \\
x_{2} \left (t \right ) &= {\mathrm e}^{2 t} \left (\sin \left (3 t \right ) c_3 t +\cos \left (3 t \right ) c_4 t +\sin \left (3 t \right ) c_1 +\cos \left (3 t \right ) c_2 \right ) \\
x_{3} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (\sin \left (3 t \right ) c_3 -\sin \left (3 t \right ) c_4 +\cos \left (3 t \right ) c_3 +\cos \left (3 t \right ) c_4 \right )}{6} \\
x_{4} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (3 \sin \left (3 t \right ) c_3 t +3 \sin \left (3 t \right ) c_4 t -3 \cos \left (3 t \right ) c_3 t +3 \cos \left (3 t \right ) c_4 t +3 \sin \left (3 t \right ) c_1 +3 \sin \left (3 t \right ) c_2 -9 \sin \left (3 t \right ) c_3 +10 \sin \left (3 t \right ) c_4 -3 \cos \left (3 t \right ) c_1 +3 \cos \left (3 t \right ) c_2 -10 \cos \left (3 t \right ) c_3 -9 \cos \left (3 t \right ) c_4 \right )}{18} \\
\end{align*}
✓ Mathematica. Time used: 0.025 (sec). Leaf size: 482
ode={D[ x1[t],t]==2*x1[t]+0*x2[t]-8*x3[t]-3*x4[t],D[ x2[t],t]==-18*x1[t]-1*x2[t]+0*x3[t]+0*x4[t],D[ x3[t],t]==-9*x1[t]-3*x2[t]-25*x3[t]-9*x4[t],D[ x4[t],t]==33*x1[t]+10*x2[t]+90*x3[t]+32*x4[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to \frac {1}{2} e^{(2-3 i) t} \left (c_1 \left (e^{6 i t} (1+3 i t)-3 i t+1\right )+i (3 c_3+c_4) \left (-1+e^{6 i t}\right )+t \left (i c_2 \left (-1+e^{6 i t}\right )+c_3 \left ((1+9 i) e^{6 i t}+(1-9 i)\right )+3 i c_4 \left (-1+e^{6 i t}\right )\right )\right ) \\
\text {x2}(t)\to -\frac {1}{2} e^{(2-3 i) t} \left (c_1 \left ((9-9 i) t+e^{6 i t} ((9+9 i) t-3 i)+3 i\right )+c_2 \left ((3-3 i) t+e^{6 i t} (-1+(3+3 i) t)-1\right )+10 i c_3 e^{6 i t}+(30+24 i) c_3 e^{6 i t} t+(30-24 i) c_3 t+3 i c_4 e^{6 i t}+(9+9 i) c_4 e^{6 i t} t+(9-9 i) c_4 t-10 i c_3-3 i c_4\right ) \\
\text {x3}(t)\to \frac {1}{2} e^{(2-3 i) t} \left (3 i c_1 \left (-1+e^{6 i t}\right )+i c_2 \left (-1+e^{6 i t}\right )+(1+9 i) c_3 e^{6 i t}+3 i c_4 e^{6 i t}+(1-9 i) c_3-3 i c_4\right ) \\
\text {x4}(t)\to \frac {1}{2} e^{(2-3 i) t} \left (c_1 \left (3 t+e^{6 i t} (3 t-10 i)+10 i\right )+c_2 \left (t+e^{6 i t} (t-3 i)+3 i\right )-27 i c_3 e^{6 i t}+(9-i) c_3 e^{6 i t} t+(9+i) c_3 t+(1-9 i) c_4 e^{6 i t}+3 c_4 e^{6 i t} t+3 c_4 t+27 i c_3+(1+9 i) c_4\right ) \\
\end{align*}
✓ Sympy. Time used: 0.316 (sec). Leaf size: 224
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
x__3 = Function("x__3")
x__4 = Function("x__4")
ode=[Eq(-2*x__1(t) + 8*x__3(t) + 3*x__4(t) + Derivative(x__1(t), t),0),Eq(18*x__1(t) + x__2(t) + Derivative(x__2(t), t),0),Eq(9*x__1(t) + 3*x__2(t) + 25*x__3(t) + 9*x__4(t) + Derivative(x__3(t), t),0),Eq(-33*x__1(t) - 10*x__2(t) - 90*x__3(t) - 32*x__4(t) + Derivative(x__4(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - C_{1} t e^{2 t} \sin {\left (3 t \right )} + C_{2} t e^{2 t} \cos {\left (3 t \right )} - \left (3 C_{1} - C_{4}\right ) e^{2 t} \cos {\left (3 t \right )} - \left (3 C_{2} + C_{3}\right ) e^{2 t} \sin {\left (3 t \right )}, \ x^{2}{\left (t \right )} = t \left (3 C_{1} - 3 C_{2}\right ) e^{2 t} \sin {\left (3 t \right )} - t \left (3 C_{1} + 3 C_{2}\right ) e^{2 t} \cos {\left (3 t \right )} - \left (- 10 C_{1} + 9 C_{2} + 3 C_{3} + 3 C_{4}\right ) e^{2 t} \cos {\left (3 t \right )} + \left (9 C_{1} + 10 C_{2} + 3 C_{3} - 3 C_{4}\right ) e^{2 t} \sin {\left (3 t \right )}, \ x^{3}{\left (t \right )} = - C_{1} e^{2 t} \sin {\left (3 t \right )} + C_{2} e^{2 t} \cos {\left (3 t \right )}, \ x^{4}{\left (t \right )} = C_{1} t e^{2 t} \cos {\left (3 t \right )} + C_{2} t e^{2 t} \sin {\left (3 t \right )} + C_{3} e^{2 t} \cos {\left (3 t \right )} + C_{4} e^{2 t} \sin {\left (3 t \right )}\right ]
\]