76.29.2 problem 2

Internal problem ID [17878]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.7 (Defective Matrices). Problems at page 444
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 11:09:37 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.061 (sec). Leaf size: 23

dsolve([diff(x__1(t),t)=3*x__1(t)-9*x__2(t),diff(x__2(t),t)=1*x__1(t)-3*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 114

DSolve[{D[x1[t],t]==3*x1[t]-9*x2[t],D[x2[t],t]==1*x1[t]-2*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{11} e^{t/2} \left (11 c_1 \cos \left (\frac {\sqrt {11} t}{2}\right )+\sqrt {11} (5 c_1-18 c_2) \sin \left (\frac {\sqrt {11} t}{2}\right )\right ) \\ \text {x2}(t)\to \frac {1}{11} e^{t/2} \left (11 c_2 \cos \left (\frac {\sqrt {11} t}{2}\right )+\sqrt {11} (2 c_1-5 c_2) \sin \left (\frac {\sqrt {11} t}{2}\right )\right ) \\ \end{align*}