20.19.9 problem 10

Internal problem ID [3889]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.8 (Matrix exponential function), page 642
Problem number : 10
Date solved : Monday, January 27, 2025 at 08:04:05 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=6 x_{2} \left (t \right )-7 x_{3} \left (t \right )+3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{3} \left (t \right )-x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-4 x_{2} \left (t \right )+9 x_{3} \left (t \right )-3 x_{4} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 82

dsolve([diff(x__1(t),t)=1*x__1(t)+0*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__2(t),t)=0*x__1(t)+6*x__2(t)-7*x__3(t)+3*x__4(t),diff(x__3(t),t)=0*x__1(t)+0*x__2(t)+3*x__3(t)-x__4(t),diff(x__4(t),t)=0*x__1(t)-4*x__2(t)+9*x__3(t)-3*x__4(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 252

DSolve[{D[x1[t],t]==1*x1[t]+0*x2[t]+0*x3[t]+0*x4[t],D[x2[t],t]==0*x1[t]+6*x2[t]-7*x3[t]+3*x4[t],D[x3[t],t]==0*x1[t]+0*x2[t]+3*x3[t]-x4[t],D[x4[t],t]==0*x1[t]-4*x2[t]+9*x3[t]-3*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 e^t \\ \text {x2}(t)\to e^{2 t} \left (c_2 \left (2 t^2+4 t+1\right )-c_3 t (4 t+7)+c_4 t (2 t+3)\right ) \\ \text {x3}(t)\to e^{2 t} \left (2 (c_2-2 c_3+c_4) t^2+(c_3-c_4) t+c_3\right ) \\ \text {x4}(t)\to e^{2 t} \left (2 (c_2-2 c_3+c_4) t^2+(-4 c_2+9 c_3-5 c_4) t+c_4\right ) \\ \text {x1}(t)\to 0 \\ \text {x2}(t)\to e^{2 t} \left (c_2 \left (2 t^2+4 t+1\right )-c_3 t (4 t+7)+c_4 t (2 t+3)\right ) \\ \text {x3}(t)\to e^{2 t} \left (2 (c_2-2 c_3+c_4) t^2+(c_3-c_4) t+c_3\right ) \\ \text {x4}(t)\to e^{2 t} \left (2 (c_2-2 c_3+c_4) t^2+(-4 c_2+9 c_3-5 c_4) t+c_4\right ) \\ \end{align*}