20.20.23 problem 23

Internal problem ID [3913]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 23
Date solved : Tuesday, March 04, 2025 at 05:19:19 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )+13 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{3} \left (t \right )+4 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=2 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.057 (sec). Leaf size: 72

ode:=[diff(x__1(t),t) = 2*x__1(t)+13*x__2(t), diff(x__2(t),t) = -x__1(t)-2*x__2(t), diff(x__3(t),t) = 2*x__3(t)+4*x__4(t), diff(x__4(t),t) = 2*x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \sin \left (3 t \right ) c_{1} +\cos \left (3 t \right ) c_{2} \\ x_{2} \left (t \right ) &= \frac {3 \cos \left (3 t \right ) c_{1}}{13}-\frac {3 \sin \left (3 t \right ) c_{2}}{13}-\frac {2 \sin \left (3 t \right ) c_{1}}{13}-\frac {2 \cos \left (3 t \right ) c_{2}}{13} \\ x_{3} \left (t \right ) &= \left (4 c_4 t +c_3 \right ) {\mathrm e}^{2 t} \\ x_{4} \left (t \right ) &= c_4 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 84

ode={D[x1[t],t]==2*x1[t]+13*x2[t]-0*x3[t]+0*x4[t],D[x2[t],t]==-1*x1[t]-2*x2[t]-0*x3[t]+0*x4[t],D[x3[t],t]==0*x1[t]+0*x2[t]+2*x3[t]+4*x4[t],D[x4[t],t]==0*x1[t]+0*x2[t]+0*x3[t]+2*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (3 t)+\frac {1}{3} (2 c_1+13 c_2) \sin (3 t) \\ \text {x2}(t)\to c_2 \cos (3 t)-\frac {1}{3} (c_1+2 c_2) \sin (3 t) \\ \text {x3}(t)\to e^{2 t} (4 c_4 t+c_3) \\ \text {x4}(t)\to c_4 e^{2 t} \\ \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 70

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) - 13*x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) + 2*x__2(t) + Derivative(x__2(t), t),0),Eq(-2*x__3(t) - 4*x__4(t) + Derivative(x__3(t), t),0),Eq(-2*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)
 

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