7.25.3 problem 3

Internal problem ID [623]
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 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 3
Date solved : Monday, January 27, 2025 at 02:56:27 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )+4 x_{2} \left (t \right )\\ x_{2}^{\prime }\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 ) = 1\\ x_{2} \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 33

dsolve([diff(x__1(t),t) = 3*x__1(t)+4*x__2(t), diff(x__2(t),t) = 3*x__1(t)+2*x__2(t), x__1(0) = 1, x__2(0) = 1], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {8 \,{\mathrm e}^{6 t}}{7}-\frac {{\mathrm e}^{-t}}{7} \\ x_{2} \left (t \right ) &= \frac {6 \,{\mathrm e}^{6 t}}{7}+\frac {{\mathrm e}^{-t}}{7} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 44

DSolve[{D[x1[t],t]==3*x1[t]+4*x2[t],D[x2[t],t]==3*x1[t]+2*x2[t]},{x1[0]==1,x2[0]==1},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{7} e^{-t} \left (8 e^{7 t}-1\right ) \\ \text {x2}(t)\to \frac {1}{7} e^{-t} \left (6 e^{7 t}+1\right ) \\ \end{align*}