9.4.1 problem problem 1

Internal problem ID [965]
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 1
Date solved : Monday, January 27, 2025 at 03:22:38 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 34

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

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}+c_2 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}-c_2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 68

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

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (e^{4 t}+1\right )+c_2 \left (e^{4 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (e^{4 t}-1\right )+c_2 \left (e^{4 t}+1\right )\right ) \\ \end{align*}