9.4.3 problem problem 3

Internal problem ID [967]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 3
Date solved : Monday, January 27, 2025 at 03:22:39 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 )&=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: 0.006 (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*}