14.32.3 problem 3

Internal problem ID [2827]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.7 (Phase portraits of linear systems). Page 427
Problem number : 3
Date solved : Monday, January 27, 2025 at 06:19:03 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-2 x_{1} \left (t \right )+5 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 34

dsolve([diff(x__1(t),t)=4*x__1(t)-x__2(t),diff(x__2(t),t)=-2*x__1(t)+5*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}+c_2 \,{\mathrm e}^{6 t} \\ x_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}-2 c_2 \,{\mathrm e}^{6 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 72

DSolve[{D[x1[t],t]==4*x1[t]-x2[t],D[x2[t],t]==-2*x1[t]+5*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{3 t} \left (c_1 \left (e^{3 t}+2\right )-c_2 \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{3 t} \left (c_2 \left (2 e^{3 t}+1\right )-2 c_1 \left (e^{3 t}-1\right )\right ) \\ \end{align*}