9.6.33 problem problem 33

Internal problem ID [1040]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 33
Date solved : Saturday, March 29, 2025 at 10:37:31 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )-4 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )+3 x_{2} \left (t \right )+x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{3} \left (t \right )-4 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=4 x_{3} \left (t \right )+3 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.658 (sec). Leaf size: 137

ode:=[diff(x__1(t),t) = 3*x__1(t)-4*x__2(t)+x__3(t), diff(x__2(t),t) = 4*x__1(t)+3*x__2(t)+x__4(t), diff(x__3(t),t) = 3*x__3(t)-4*x__4(t), diff(x__4(t),t) = 4*x__3(t)+3*x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (4 \sin \left (4 t \right ) c_3 t +4 \cos \left (4 t \right ) c_4 t +4 \sin \left (4 t \right ) c_2 -\sin \left (4 t \right ) c_4 +4 \cos \left (4 t \right ) c_1 \right )}{4} \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (4 \sin \left (4 t \right ) c_4 t -4 \cos \left (4 t \right ) c_3 t +4 \sin \left (4 t \right ) c_1 -4 \cos \left (4 t \right ) c_2 +c_4 \cos \left (4 t \right )\right )}{4} \\ x_{3} \left (t \right ) &= {\mathrm e}^{3 t} \left (c_3 \sin \left (4 t \right )+c_4 \cos \left (4 t \right )\right ) \\ x_{4} \left (t \right ) &= {\mathrm e}^{3 t} \left (\sin \left (4 t \right ) c_4 -\cos \left (4 t \right ) c_3 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.092 (sec). Leaf size: 120

ode={D[ x1[t],t]==3*x1[t]-4*x2[t]+1*x3[t]+0*x4[t],D[ x2[t],t]==4*x1[t]+3*x2[t]+0*x3[t]+1*x4[t],D[ x3[t],t]==0*x1[t]+0*x2[t]+3*x3[t]-4*x4[t],D[ x4[t],t]==0*x1[t]+0*x2[t]+4*x3[t]+3*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{3 t} ((c_3 t+c_1) \cos (4 t)-(c_4 t+c_2) \sin (4 t)) \\ \text {x2}(t)\to e^{3 t} ((c_4 t+c_2) \cos (4 t)+(c_3 t+c_1) \sin (4 t)) \\ \text {x3}(t)\to e^{3 t} (c_3 \cos (4 t)-c_4 \sin (4 t)) \\ \text {x4}(t)\to e^{3 t} (c_4 \cos (4 t)+c_3 \sin (4 t)) \\ \end{align*}

✓ Sympy. Time used: 0.222 (sec). Leaf size: 160

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(-3*x__1(t) + 4*x__2(t) - x__3(t) + Derivative(x__1(t), t),0),Eq(-4*x__1(t) - 3*x__2(t) - x__4(t) + Derivative(x__2(t), t),0),Eq(-3*x__3(t) + 4*x__4(t) + Derivative(x__3(t), t),0),Eq(-4*x__3(t) - 3*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)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{1} t e^{3 t} \sin {\left (4 t \right )} - C_{2} t e^{3 t} \cos {\left (4 t \right )} - C_{3} e^{3 t} \sin {\left (4 t \right )} - C_{4} e^{3 t} \cos {\left (4 t \right )}, \ x^{2}{\left (t \right )} = C_{1} t e^{3 t} \cos {\left (4 t \right )} - C_{2} t e^{3 t} \sin {\left (4 t \right )} + C_{3} e^{3 t} \cos {\left (4 t \right )} - C_{4} e^{3 t} \sin {\left (4 t \right )}, \ x^{3}{\left (t \right )} = - C_{1} e^{3 t} \sin {\left (4 t \right )} - C_{2} e^{3 t} \cos {\left (4 t \right )}, \ x^{4}{\left (t \right )} = C_{1} e^{3 t} \cos {\left (4 t \right )} - C_{2} e^{3 t} \sin {\left (4 t \right )}\right ] \]