20.16.4 problem 4

Internal problem ID [3837]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.4 (Nondefective coefficient matrix), page 607
Problem number : 4
Date solved : Monday, January 27, 2025 at 08:03:13 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 47

dsolve([diff(x__1(t),t)=x__1(t)-2*x__2(t),diff(x__2(t),t)=5*x__1(t)-5*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 56

DSolve[{D[x1[t],t]==x1[t]-2*x2[t],D[x2[t],t]==5*x1[t]-5*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-2 t} (c_1 \cos (t)+(3 c_1-2 c_2) \sin (t)) \\ \text {x2}(t)\to e^{-2 t} (c_2 \cos (t)+(5 c_1-3 c_2) \sin (t)) \\ \end{align*}