20.18.1 problem 1

Internal problem ID [3871]
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 : 1
Date solved : Monday, January 27, 2025 at 08:03:41 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 65

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

\begin{align*} x_{1} \left (t \right ) &= \left (c_{2} +3 t \right ) {\mathrm e}^{t}+{\mathrm e}^{2 t} c_{1} +3 \,{\mathrm e}^{2 t} t -3 \,{\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= \left (c_{2} +3 t -1\right ) {\mathrm e}^{t}+\frac {2 \,{\mathrm e}^{2 t} c_{1}}{3}+2 \,{\mathrm e}^{2 t} t -\frac {8 \,{\mathrm e}^{2 t}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 73

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

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