7.25.24 problem 24

Internal problem ID [644]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 24
Date solved : Wednesday, February 05, 2025 at 03:51:02 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 86

dsolve([diff(x__1(t),t)=2*x__1(t)+x__2(t)-x__3(t),diff(x__2(t),t)=-4*x__1(t)-3*x__2(t)-x__3(t),diff(x__3(t),t)=4*x__1(t)+4*x__2(t)+2*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 103

DSolve[{D[x1[t],t]==2*x1[t]+x2[t]-x3[t],D[x2[t],t]==-4*x1[t]-3*x2[t]-x3[t],D[x3[t],t]==4*x1[t]+4*x2[t]+2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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