20.20.24 problem 24

Internal problem ID [3914]
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 : 24
Date solved : Tuesday, March 04, 2025 at 05:19:20 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=7 x_{1} \left (t \right )-x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=6 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{3} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=2 x_{1} \left (t \right )+5 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 62

ode:=[diff(x__1(t),t) = 7*x__1(t)-x__4(t), diff(x__2(t),t) = 6*x__2(t), diff(x__3(t),t) = -x__3(t), diff(x__4(t),t) = 2*x__1(t)+5*x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{6 t} \left (c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= c_4 \,{\mathrm e}^{6 t} \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{-t} \\ x_{4} \left (t \right ) &= {\mathrm e}^{6 t} \left (c_{1} \sin \left (t \right )+\sin \left (t \right ) c_{2} -\cos \left (t \right ) c_{1} +c_{2} \cos \left (t \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 267

ode={D[x1[t],t]==7*x1[t]+0*x2[t]-0*x3[t]-1*x4[t],D[x2[t],t]==0*x1[t]+6*x2[t]-0*x3[t]+0*x4[t],D[x3[t],t]==0*x1[t]+0*x2[t]-1*x3[t]+0*x4[t],D[x4[t],t]==2*x1[t]+0*x2[t]+0*x3[t]+5*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{6 t} (c_1 (\sin (t)+\cos (t))-c_2 \sin (t)) \\ \text {x4}(t)\to e^{6 t} (c_2 \cos (t)+(2 c_1-c_2) \sin (t)) \\ \text {x2}(t)\to c_3 e^{6 t} \\ \text {x3}(t)\to c_4 e^{-t} \\ \text {x1}(t)\to e^{6 t} (c_1 (\sin (t)+\cos (t))-c_2 \sin (t)) \\ \text {x4}(t)\to e^{6 t} (c_2 \cos (t)+(2 c_1-c_2) \sin (t)) \\ \text {x2}(t)\to c_3 e^{6 t} \\ \text {x3}(t)\to 0 \\ \text {x1}(t)\to e^{6 t} (c_1 (\sin (t)+\cos (t))-c_2 \sin (t)) \\ \text {x4}(t)\to e^{6 t} (c_2 \cos (t)+(2 c_1-c_2) \sin (t)) \\ \text {x2}(t)\to 0 \\ \text {x3}(t)\to c_4 e^{-t} \\ \text {x1}(t)\to e^{6 t} (c_1 (\sin (t)+\cos (t))-c_2 \sin (t)) \\ \text {x4}(t)\to e^{6 t} (c_2 \cos (t)+(2 c_1-c_2) \sin (t)) \\ \text {x2}(t)\to 0 \\ \text {x3}(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.137 (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(-7*x__1(t) + x__4(t) + Derivative(x__1(t), t),0),Eq(-6*x__2(t) + Derivative(x__2(t), t),0),Eq(x__3(t) + Derivative(x__3(t), t),0),Eq(-2*x__1(t) - 5*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)
 

\[ \left [ x^{1}{\left (t \right )} = \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{6 t} \cos {\left (t \right )} - \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{6 t} \sin {\left (t \right )}, \ x^{2}{\left (t \right )} = C_{3} e^{6 t}, \ x^{3}{\left (t \right )} = C_{4} e^{- t}, \ x^{4}{\left (t \right )} = C_{1} e^{6 t} \cos {\left (t \right )} - C_{2} e^{6 t} \sin {\left (t \right )}\right ] \]