7.24.15 problem 25 and 34

Internal problem ID [615]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.3 (Matrices and linear systems). Problems at page 364
Problem number : 25 and 34
Date solved : Monday, January 27, 2025 at 02:56:23 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-3 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=6 x_{1} \left (t \right )-7 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 8\\ x_{2} \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 33

dsolve([diff(x__1(t),t) = 4*x__1(t)-3*x__2(t), diff(x__2(t),t) = 6*x__1(t)-7*x__2(t), x__1(0) = 8, x__2(0) = 0], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {16 \,{\mathrm e}^{-5 t}}{7}+\frac {72 \,{\mathrm e}^{2 t}}{7} \\ x_{2} \left (t \right ) &= -\frac {48 \,{\mathrm e}^{-5 t}}{7}+\frac {48 \,{\mathrm e}^{2 t}}{7} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 42

DSolve[{D[x1[t],t]==4*x1[t]-3*x2[t],D[x2[t],t]==6*x1[t]-7*x2[t]},{x1[0]==8,x2[0]==0},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {8}{7} e^{-5 t} \left (9 e^{7 t}-2\right ) \\ \text {x2}(t)\to \frac {48}{7} e^{-5 t} \left (e^{7 t}-1\right ) \\ \end{align*}