20.13.13 problem 13

Internal problem ID [3822]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 13
Date solved : Monday, January 27, 2025 at 08:03:02 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 )+t\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )+x_{2} \left (t \right )+1 \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 44

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

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

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 79

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

\begin{align*} \text {x1}(t)\to -\frac {t^2}{2}+3 t+c_1 \left (2 e^{-t}-1\right )-c_2 e^{-t}-3+c_2 \\ \text {x2}(t)\to e^{-t} \left (-e^t \left (t^2-4 t+3+2 c_1-2 c_2\right )+2 c_1-c_2\right ) \\ \end{align*}