9.6.31 problem problem 31
Internal
problem
ID
[1038]
Book
:
Differential
equations
and
linear
algebra,
4th
ed.,
Edwards
and
Penney
Section
:
Section
7.6,
Multiple
Eigenvalue
Solutions.
Page
451
Problem
number
:
problem
31
Date
solved
:
Saturday, March 29, 2025 at 10:37:27 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=35 x_{1} \left (t \right )-12 x_{2} \left (t \right )+4 x_{3} \left (t \right )+30 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=22 x_{1} \left (t \right )-8 x_{2} \left (t \right )+3 x_{3} \left (t \right )+19 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-10 x_{1} \left (t \right )+3 x_{2} \left (t \right )-9 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-27 x_{1} \left (t \right )+9 x_{2} \left (t \right )-3 x_{3} \left (t \right )-23 x_{4} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.184 (sec). Leaf size: 116
ode:=[diff(x__1(t),t) = 35*x__1(t)-12*x__2(t)+4*x__3(t)+30*x__4(t), diff(x__2(t),t) = 22*x__1(t)-8*x__2(t)+3*x__3(t)+19*x__4(t), diff(x__3(t),t) = -10*x__1(t)+3*x__2(t)-9*x__4(t), diff(x__4(t),t) = -27*x__1(t)+9*x__2(t)-3*x__3(t)-23*x__4(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-6 c_4 \,t^{2}-6 c_3 t -4 c_4 t +3 c_1 -6 c_2 -2 c_3 \right )}{3} \\
x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-3 c_4 \,t^{2}-3 c_3 t -16 c_4 t +3 c_1 -3 c_2 -8 c_3 +6 c_4 \right )}{9} \\
x_{3} \left (t \right ) &= {\mathrm e}^{t} \left (c_4 \,t^{2}+c_3 t +c_2 \right ) \\
x_{4} \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (-18 c_4 \,t^{2}-18 c_3 t -6 c_4 t +9 c_1 -18 c_2 -3 c_3 -2 c_4 \right )}{9} \\
\end{align*}
✓ Mathematica. Time used: 0.005 (sec). Leaf size: 207
ode={D[ x1[t],t]==35*x1[t]-12*x2[t]+4*x3[t]+30*x4[t],D[ x2[t],t]==22*x1[t]-8*x2[t]+3*x3[t]+19*x4[t],D[ x3[t],t]==-10*x1[t]+3*x2[t]+0*x3[t]-9*x4[t],D[ x4[t],t]==-27*x1[t]+9*x2[t]-3*x3[t]-23*x4[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to e^t \left (c_1 \left (21 t^2+34 t+1\right )-3 c_2 t (3 t+4)+c_3 t (3 t+4)+6 c_4 t (3 t+5)\right ) \\
\text {x2}(t)\to \frac {1}{2} e^t \left ((7 c_1-3 c_2+c_3+6 c_4) t^2+2 (22 c_1-9 c_2+3 c_3+19 c_4) t+2 c_2\right ) \\
\text {x3}(t)\to \frac {1}{2} e^t \left (-3 (7 c_1-3 c_2+c_3+6 c_4) t^2-2 (10 c_1-3 c_2+c_3+9 c_4) t+2 c_3\right ) \\
\text {x4}(t)\to e^t \left (-3 (7 c_1-3 c_2+c_3+6 c_4) t^2-3 (9 c_1-3 c_2+c_3+8 c_4) t+c_4\right ) \\
\end{align*}
✓ Sympy. Time used: 0.237 (sec). Leaf size: 150
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(-35*x__1(t) + 12*x__2(t) - 4*x__3(t) - 30*x__4(t) + Derivative(x__1(t), t),0),Eq(-22*x__1(t) + 8*x__2(t) - 3*x__3(t) - 19*x__4(t) + Derivative(x__2(t), t),0),Eq(10*x__1(t) - 3*x__2(t) + 9*x__4(t) + Derivative(x__3(t), t),0),Eq(27*x__1(t) - 9*x__2(t) + 3*x__3(t) + 23*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)
\[
\left [ x^{1}{\left (t \right )} = 21 C_{1} t^{2} e^{t} + t \left (34 C_{1} + 42 C_{2}\right ) e^{t} + \left (C_{1} + 34 C_{2} + 42 C_{3}\right ) e^{t}, \ x^{2}{\left (t \right )} = \frac {7 C_{1} t^{2} e^{t}}{2} + t \left (22 C_{1} + 7 C_{2}\right ) e^{t} + \left (22 C_{2} + 7 C_{3} + \frac {C_{4}}{3}\right ) e^{t}, \ x^{3}{\left (t \right )} = - \frac {21 C_{1} t^{2} e^{t}}{2} - t \left (10 C_{1} + 21 C_{2}\right ) e^{t} - \left (10 C_{2} + 21 C_{3} - C_{4}\right ) e^{t}, \ x^{4}{\left (t \right )} = - 21 C_{1} t^{2} e^{t} - t \left (27 C_{1} + 42 C_{2}\right ) e^{t} - \left (27 C_{2} + 42 C_{3}\right ) e^{t}\right ]
\]