20.18.10 problem 11

Internal problem ID [3880]
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 : 11
Date solved : Monday, January 27, 2025 at 08:03:49 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-3 x_{2} \left (t \right )+34 \sin \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-4 x_{1} \left (t \right )-2 x_{2} \left (t \right )+17 \cos \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.163 (sec). Leaf size: 49

dsolve([diff(x__1(t),t)=2*x__1(t)-3*x__2(t)+34*sin(t),diff(x__2(t),t)=-4*x__1(t)-2*x__2(t)+17*cos(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{4 t}+{\mathrm e}^{-4 t} c_{1} +\cos \left (t \right )-4 \sin \left (t \right ) \\ x_{2} \left (t \right ) &= -\frac {2 c_{2} {\mathrm e}^{4 t}}{3}+2 \,{\mathrm e}^{-4 t} c_{1} +9 \sin \left (t \right )+2 \cos \left (t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 98

DSolve[{D[x1[t],t]==2*x1[t]-3*x2[t]+34*Sin[t],D[x2[t],t]==-4*x1[t]-2*x2[t]+17*Cos[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -4 \sin (t)+\cos (t)+\frac {1}{4} c_1 e^{-4 t} \left (3 e^{8 t}+1\right )-\frac {3}{8} c_2 e^{-4 t} \left (e^{8 t}-1\right ) \\ \text {x2}(t)\to 9 \sin (t)+2 \cos (t)-\frac {1}{2} c_1 e^{-4 t} \left (e^{8 t}-1\right )+\frac {1}{4} c_2 e^{-4 t} \left (e^{8 t}+3\right ) \\ \end{align*}