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 : Monday, January 27, 2025 at 08:04:27 AM
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*}

Solution by Maple

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

dsolve([diff(x__1(t),t)=7*x__1(t)+0*x__2(t)-0*x__3(t)-1*x__4(t),diff(x__2(t),t)=0*x__1(t)+6*x__2(t)-0*x__3(t)+0*x__4(t),diff(x__3(t),t)=0*x__1(t)+0*x__2(t)-1*x__3(t)+0*x__4(t),diff(x__4(t),t)=2*x__1(t)+0*x__2(t)+0*x__3(t)+5*x__4(t)],singsol=all)
 
\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*}

Solution by Mathematica

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

DSolve[{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]},{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*}