20.18.7 problem 7

Internal problem ID [3877]
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 : 7
Date solved : Monday, January 27, 2025 at 08:03:47 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )+2 x_{2} \left (t \right )-3 \,{\mathrm e}^{t}\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )-x_{2} \left (t \right )+6 t \,{\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.057 (sec). Leaf size: 55

dsolve([diff(x__1(t),t)=3*x__1(t)+2*x__2(t)-3*exp(t),diff(x__2(t),t)=-2*x__1(t)-x__2(t)+6*t*exp(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (2 t^{3}+c_{1} t -3 t^{2}+c_{2} +t \right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (4 t^{3}+2 c_{1} t -12 t^{2}-c_{1} +2 c_{2} +8 t -4\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 62

DSolve[{D[x1[t],t]==3*x1[t]+2*x2[t]-3*Exp[t],D[x2[t],t]==-2*x1[t]-x2[t]+6*t*Exp[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^t \left (2 t^3-3 t^2+(-3+2 c_1+2 c_2) t+c_1\right ) \\ \text {x2}(t)\to e^t \left (-2 t^3+6 t^2-2 (c_1+c_2) t+c_2\right ) \\ \end{align*}