14.22.2 problem 2

Internal problem ID [2729]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.8, Systems of differential equations. The eigenva1ue-eigenvector method. Page 339
Problem number : 2
Date solved : Monday, January 27, 2025 at 06:12:25 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 34

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} c_1 +c_2 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= 4 \,{\mathrm e}^{2 t} c_1 +c_2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

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

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

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-t} \left (c_2 \left (e^{3 t}-1\right )-c_1 \left (e^{3 t}-4\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-t} \left (c_2 \left (4 e^{3 t}-1\right )-4 c_1 \left (e^{3 t}-1\right )\right ) \\ \end{align*}