10.19.14 problem 14

Internal problem ID [1455]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 9.1, The Phase Plane: Linear Systems. page 505
Problem number : 14
Date solved : Monday, January 27, 2025 at 04:57:34 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.105 (sec). Leaf size: 72

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

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