20.18.4 problem 4

Internal problem ID [3874]
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 : 4
Date solved : Monday, January 27, 2025 at 08:03:43 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )+x_{2} \left (t \right )+20 \,{\mathrm e}^{3 t}\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{1} \left (t \right )+x_{2} \left (t \right )+12 \,{\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 55

dsolve([diff(x__1(t),t)=-x__1(t)+x__2(t)+20*exp(3*t),diff(x__2(t),t)=3*x__1(t)+x__2(t)+12*exp(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.819 (sec). Leaf size: 135

DSolve[{D[x1[t],t]==x1[t]+x2[t]+20*Exp[3*t],D[x2[t],t]==3*x1[t]+x2[t]+12*Exp[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{6} e^t \left (240 e^{2 t}+\left (3 c_1-\sqrt {3} c_2\right ) e^{-\sqrt {3} t}+\left (3 c_1+\sqrt {3} c_2\right ) e^{\sqrt {3} t}-24\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{t-\sqrt {3} t} \left (120 e^{\left (2+\sqrt {3}\right ) t}+\left (\sqrt {3} c_1+c_2\right ) e^{2 \sqrt {3} t}-\sqrt {3} c_1+c_2\right ) \\ \end{align*}