9.4.2 problem problem 2

Internal problem ID [966]
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 2
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 )&=2 x_{1} \left (t \right )+3 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: 0.006 (sec). Leaf size: 35

dsolve([diff(x__1(t),t)=2*x__1(t)+3*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}^{4 t}+c_2 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= \frac {2 c_1 \,{\mathrm e}^{4 t}}{3}-c_2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 74

DSolve[{D[ x1[t],t]==2*x1[t]+3*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}{5} e^{-t} \left (c_1 \left (3 e^{5 t}+2\right )+3 c_2 \left (e^{5 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{5} e^{-t} \left (2 c_1 \left (e^{5 t}-1\right )+c_2 \left (2 e^{5 t}+3\right )\right ) \\ \end{align*}