76.29.9 problem 9

Internal problem ID [17885]
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 : 9
Date solved : Tuesday, January 28, 2025 at 11:09:43 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 7\\ x_{2} \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 28

dsolve([diff(x__1(t),t) = x__1(t)-4*x__2(t), diff(x__2(t),t) = 4*x__1(t)-7*x__2(t), x__1(0) = 7, x__2(0) = 1], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (24 t +7\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (96 t +4\right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 30

DSolve[{D[x1[t],t]==1*x1[t]-4*x2[t],D[x2[t],t]==4*x1[t]-7*x2[t]},{x1[0]==7,x2[0]==1},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-3 t} (24 t+7) \\ \text {x2}(t)\to e^{-3 t} (24 t+1) \\ \end{align*}