20.14.2 problem 13

Internal problem ID [3826]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.2, page 592
Problem number : 13
Date solved : Monday, January 27, 2025 at 08:03:05 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-b x_{1} \left (t \right )-a x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 115

dsolve([diff(x__1(t),t)=x__2(t),diff(x__2(t),t)=-b*x__1(t)-a*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 227

DSolve[{D[x1[t],t]==x2[t],D[x2[t],t]==-b*x1[t]-a*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

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