76.27.16 problem 16

Internal problem ID [17862]
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 6.5 (Fundamental Matrices and the Exponential of a Matrix). Problems at page 430
Problem number : 16
Date solved : Tuesday, January 28, 2025 at 11:04:12 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 29

dsolve([diff(x__1(t),t) = 2*x__1(t)-x__2(t), diff(x__2(t),t) = 3*x__1(t)-2*x__2(t), x__1(0) = 2, x__2(0) = -1], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {7 \,{\mathrm e}^{t}}{2}-\frac {3 \,{\mathrm e}^{-t}}{2} \\ x_{2} \left (t \right ) &= \frac {7 \,{\mathrm e}^{t}}{2}-\frac {9 \,{\mathrm e}^{-t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 42

DSolve[{D[x1[t],t]==2*x1[t]-1*x2[t],D[x2[t],t]==3*x1[t]-2*x2[t]},{x1[0]==2,x2[0]==-1},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {7 e^t}{2}-\frac {3 e^{-t}}{2} \\ \text {x2}(t)\to \frac {7 e^t}{2}-\frac {9 e^{-t}}{2} \\ \end{align*}