20.18.5 problem 5

Internal problem ID [3875]
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 : 5
Date solved : Monday, January 27, 2025 at 08:03:44 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )+2 x_{2} \left (t \right )+54 t \,{\mathrm e}^{3 t}\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )+4 x_{2} \left (t \right )+9 \,{\mathrm e}^{3 t} \end{align*}

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 60

dsolve([diff(x__1(t),t)=-x__1(t)+2*x__2(t)+54*t*exp(3*t),diff(x__2(t),t)=-2*x__1(t)+4*x__2(t)+9*exp(3*t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \left (-9 t^{2}+\frac {1}{3} c_{1} +30 t -10\right ) {\mathrm e}^{3 t}+c_{2} \\ x_{2} \left (t \right ) &= 24 \,{\mathrm e}^{3 t} t -18 \,{\mathrm e}^{3 t} t^{2}+\frac {2 c_{1} {\mathrm e}^{3 t}}{3}-5 \,{\mathrm e}^{3 t}+\frac {c_{2}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.075 (sec). Leaf size: 81

DSolve[{D[x1[t],t]==-x1[t]+2*x2[t]+54*t*Exp[3*t],D[x2[t],t]==-2*x1[t]+4*x2[t]+9*Exp[3*t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} \left (-e^{3 t} \left (27 t^2-90 t+30+c_1-2 c_2\right )+4 c_1-2 c_2\right ) \\ \text {x2}(t)\to \frac {1}{3} \left (e^{3 t} \left (-54 t^2+72 t-15-2 c_1+4 c_2\right )+2 c_1-c_2\right ) \\ \end{align*}