76.28.4 problem 5

Internal problem ID [17871]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.6 (Nonhomogeneous Linear Systems). Problems at page 436
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 11:04:19 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 )&=4 x_{1} \left (t \right )-2 x_{2} \left (t \right )-2 \,{\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.098 (sec). Leaf size: 44

dsolve([diff(x__1(t),t)=1*x__1(t)+1*x__2(t)+exp(-2*t),diff(x__2(t),t)=4*x__1(t)-2*x__2(t)-2*exp(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.615 (sec). Leaf size: 84

DSolve[{D[x1[t],t]==1*x1[t]+1*x2[t]+Exp[-2*t],D[x2[t],t]==4*x1[t]-2*x2[t]-2*Exp[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {e^t}{2}+\frac {1}{5} (c_1-c_2) e^{-3 t}+\frac {1}{5} (4 c_1+c_2) e^{2 t} \\ \text {x2}(t)\to \frac {1}{5} e^{-3 t} \left (-5 e^t+(4 c_1+c_2) e^{5 t}-4 c_1+4 c_2\right ) \\ \end{align*}