20.13.12 problem 12

Internal problem ID [3821]
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 : 12
Date solved : Monday, January 27, 2025 at 08:03:01 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 )+5 \,{\mathrm e}^{4 t}\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 46

dsolve([diff(x__1(t),t)=x__1(t)+2*x__2(t)+5*exp(4*t),diff(x__2(t),t)=2*x__1(t)+x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 76

DSolve[{D[x1[t],t]==x1[t]+2*x2[t]+5*Exp[4*t],D[x2[t],t]==2*x1[t]+x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-t} \left (6 e^{5 t}+(c_1+c_2) e^{4 t}+c_1-c_2\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (4 e^{5 t}+(c_1+c_2) e^{4 t}-c_1+c_2\right ) \\ \end{align*}