20.13.7 problem 7

Internal problem ID [3816]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 7
Date solved : Monday, January 27, 2025 at 08:02:57 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 41

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

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

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 92

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

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