20.13.14 problem 14

Internal problem ID [3823]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 14
Date solved : Monday, January 27, 2025 at 08:03:03 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 52

dsolve([diff(x__1(t),t)=x__1(t)+x__2(t)+exp(2*t),diff(x__2(t),t)=3*x__1(t)-x__2(t)+5*exp(2*t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 76

DSolve[{D[x1[t],t]==x1[t]+x2[t]+Exp[2*t],D[x2[t],t]==3*x1[t]-x2[t]+5*Exp[2*t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-2 t} \left (e^{4 t} (8 t-1+3 c_1+c_2)+c_1-c_2\right ) \\ \text {x2}(t)\to \frac {1}{4} e^{-2 t} \left (e^{4 t} (8 t+3+3 c_1+c_2)-3 c_1+3 c_2\right ) \\ \end{align*}