14.22.8 problem 8

Internal problem ID [2735]
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 : 8
Date solved : Monday, January 27, 2025 at 06:12:29 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 5 \end{align*}

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 33

dsolve([diff(x__1(t),t) = x__1(t)-3*x__2(t), diff(x__2(t),t) = -2*x__1(t)+2*x__2(t), x__1(0) = 0, x__2(0) = 5], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 37

DSolve[{D[ x1[t],t]==1*x1[t]-3*x2[t],D[ x2[t],t]==-2*x1[t]+2*x2[t]},{x1[0]==0,x2[0]==5},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -3 e^{-t} \left (e^{5 t}-1\right ) \\ \text {x2}(t)\to e^{-t} \left (3 e^{5 t}+2\right ) \\ \end{align*}