20.18.3 problem 3

Internal problem ID [3873]
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 : 3
Date solved : Monday, January 27, 2025 at 08:03:43 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 34

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

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