48.5.6 problem Problem 5.7

Internal problem ID [7571]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 5. Systems of First Order Differential Equations. Section 5.11 Problems. Page 360
Problem number : Problem 5.7
Date solved : Monday, January 27, 2025 at 03:06:25 PM
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 \,{\mathrm e}^{-t}\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+3 t \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 64

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

\begin{align*} x_{1} \left (t \right ) &= -c_{2} {\mathrm e}^{-3 t}+{\mathrm e}^{-t} c_{1} +\frac {{\mathrm e}^{-t}}{2}-\frac {4}{3}+{\mathrm e}^{-t} t +t \\ x_{2} \left (t \right ) &= c_{2} {\mathrm e}^{-3 t}+{\mathrm e}^{-t} c_{1} -\frac {{\mathrm e}^{-t}}{2}-\frac {5}{3}+2 t +{\mathrm e}^{-t} t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.098 (sec). Leaf size: 93

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

\begin{align*} \text {x1}(t)\to \frac {1}{6} \left (6 t+3 (c_1-c_2) e^{-3 t}+3 e^{-t} (2 t+1+c_1+c_2)-8\right ) \\ \text {x2}(t)\to \frac {1}{6} e^{-3 t} \left (2 e^{3 t} (6 t-5)+3 e^{2 t} (2 t-1+c_1+c_2)-3 c_1+3 c_2\right ) \\ \end{align*}