76.8.17 problem 17

Internal problem ID [17506]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.4 (Complex Eigenvalues). Problems at page 177
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 10:39:40 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+a y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )-y \left (t \right ) \end{align*}

Solution by Maple

Time used: 0.099 (sec). Leaf size: 71

dsolve([diff(x(t),t)=-x(t)+a*y(t),diff(y(t),t)=-x(t)-y(t)],singsol=all)
 
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\left (-1+\sqrt {-a}\right ) t}+c_{2} {\mathrm e}^{-t \sqrt {-a}-t} \\ y \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{\left (-1+\sqrt {-a}\right ) t}-c_{2} {\mathrm e}^{-t \sqrt {-a}-t}}{\sqrt {-a}} \\ \end{align*}

Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 144

DSolve[{D[x[t],t]==-x[t]+a*y[t],D[y[t],t]==-x[t]-y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 
\begin{align*} x(t)\to \frac {e^{-\left (\left (\sqrt {-a}+1\right ) t\right )} \left (\sqrt {-a} c_1 \left (e^{2 \sqrt {-a} t}+1\right )+a c_2 \left (e^{2 \sqrt {-a} t}-1\right )\right )}{2 \sqrt {-a}} \\ y(t)\to \frac {e^{-\left (\left (\sqrt {-a}+1\right ) t\right )} \left (\sqrt {-a} c_2 \left (e^{2 \sqrt {-a} t}+1\right )-c_1 \left (e^{2 \sqrt {-a} t}-1\right )\right )}{2 \sqrt {-a}} \\ \end{align*}