20.18.9 problem 9

Internal problem ID [3879]
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 : 9
Date solved : Monday, January 27, 2025 at 08:03:48 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.068 (sec). Leaf size: 65

dsolve([diff(x__1(t),t)=-x__1(t)-2*x__2(t)+2*x__3(t)-exp(3*t),diff(x__2(t),t)=2*x__1(t)+4*x__2(t)-x__3(t)+4*exp(3*t),diff(x__3(t),t)=3*x__3(t)+3*exp(3*t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \left (\frac {c_{1}}{3}-\frac {t}{3}+\frac {1}{9}\right ) {\mathrm e}^{3 t}+c_{2} \\ x_{2} \left (t \right ) &= -\frac {2 c_{1} {\mathrm e}^{3 t}}{3}+\frac {11 \,{\mathrm e}^{3 t} t}{3}-\frac {5 \,{\mathrm e}^{3 t}}{9}-\frac {c_{2}}{2}+c_3 \,{\mathrm e}^{3 t} \\ x_{3} \left (t \right ) &= \left (3 t +c_3 \right ) {\mathrm e}^{3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 106

DSolve[{D[x1[t],t]==-x1[t]-2*x2[t]+2*x3[t]-Exp[3*t],D[x2[t],t]==2*x1[t]+4*x2[t]-x3[t]+4*Exp[3*t],D[x3[t],t]==3*x3[t]+3*Exp[3*t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{9} \left (6 (2 c_1+c_2-c_3)-e^{3 t} (3 t+2+3 c_1+6 c_2-6 c_3)\right ) \\ \text {x2}(t)\to \frac {1}{9} \left (e^{3 t} (33 t+1+6 c_1+12 c_2-3 c_3)-3 (2 c_1+c_2-c_3)\right ) \\ \text {x3}(t)\to e^{3 t} (3 t+c_3) \\ \end{align*}