20.18.2 problem 2

Internal problem ID [3872]
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 : 2
Date solved : Monday, January 27, 2025 at 08:03:42 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 44

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

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

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 74

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

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