20.18.8 problem 8

Internal problem ID [3878]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.6 (Variation of parameters for linear systems), page 624
Problem number : 8
Date solved : Monday, January 27, 2025 at 08:03:47 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-{\mathrm e}^{t}\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-3 x_{2} \left (t \right )+2 x_{3} \left (t \right )+6 \,{\mathrm e}^{-t}\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+2 x_{3} \left (t \right )+{\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.093 (sec). Leaf size: 71

dsolve([diff(x__1(t),t)=x__1(t)-exp(t),diff(x__2(t),t)=2*x__1(t)-3*x__2(t)+2*x__3(t)+6*exp(-t),diff(x__3(t),t)=x__1(t)-2*x__2(t)+2*x__3(t)+exp(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \left (-t +c_3 \right ) {\mathrm e}^{t} \\ x_{2} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{t}}{2}+\frac {c_3 \,{\mathrm e}^{t}}{2}-\frac {{\mathrm e}^{t}}{6}+9 \,{\mathrm e}^{-t}+2 c_{1} {\mathrm e}^{-2 t} \\ x_{3} \left (t \right ) &= c_{2} {\mathrm e}^{t}+c_{1} {\mathrm e}^{-2 t}+t \,{\mathrm e}^{t}-\frac {{\mathrm e}^{t}}{3}+6 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.139 (sec). Leaf size: 113

DSolve[{D[x1[t],t]==x1[t]-Exp[t],D[x2[t],t]==2*x1[t]-3*x2[t]+2*x3[t]+6*Exp[-t],D[x3[t],t]==x1[t]-2*x2[t]+2*x3[t]+Exp[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^t (-t+c_1) \\ \text {x2}(t)\to \frac {1}{3} e^{-2 t} \left (27 e^t+(2 c_1-c_2+2 c_3) e^{3 t}-2 (c_1-2 c_2+c_3)\right ) \\ \text {x3}(t)\to \frac {1}{3} e^{-2 t} \left (18 e^t+e^{3 t} (3 t+c_1-2 c_2+4 c_3)-c_1+2 c_2-c_3\right ) \\ \end{align*}