76.29.10 problem 10

Internal problem ID [17886]
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 : 10
Date solved : Tuesday, January 28, 2025 at 11:09:44 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = -5\\ x_{2} \left (0\right ) = 7 \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 24

dsolve([diff(x__1(t),t) = 3*x__1(t)-4*x__2(t), diff(x__2(t),t) = x__1(t)-x__2(t), x__1(0) = -5, x__2(0) = 7], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (-38 t -5\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-76 t +28\right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 27

DSolve[{D[x1[t],t]==3*x1[t]-4*x2[t],D[x2[t],t]==1*x1[t]-1*x2[t]},{x1[0]==-5,x2[0]==7},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -e^t (38 t+5) \\ \text {x2}(t)\to e^t (7-19 t) \\ \end{align*}