28.4.40 problem 7.40

Internal problem ID [4572]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.40
Date solved : Monday, January 27, 2025 at 09:23:39 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-x_{1} \left (t \right )+8 x_{2} \left (t \right )+9 t\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+3 \,{\mathrm e}^{-t} \end{align*}

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 46

dsolve([diff(x__1(t),t)=-x__1(t)+8*x__2(t)+9*t,diff(x__2(t),t)=x__1(t)+x__2(t)+3*exp(-t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.244 (sec). Leaf size: 89

DSolve[{D[x1[t],t]==-x1[t]+8*x2[t]+9*t,D[x2[t],t]==x1[t]+x2[t]+3*Exp[-t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to t-3 e^{-t}+\frac {2}{3} (c_1-2 c_2) e^{-3 t}+\frac {1}{3} (c_1+4 c_2) e^{3 t}-1 \\ \text {x2}(t)\to \frac {1}{6} e^{-3 t} \left (-6 e^{3 t} t+(c_1+4 c_2) e^{6 t}-c_1+2 c_2\right ) \\ \end{align*}