76.8.20 problem 20

Internal problem ID [17509]
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 : 20
Date solved : Tuesday, January 28, 2025 at 10:39:43 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )+a y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=8 x \left (t \right )-6 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.095 (sec). Leaf size: 123

dsolve([diff(x(t),t)=4*x(t)+a*y(t),diff(y(t),t)=8*x(t)-6*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\left (-1+\sqrt {8 a +25}\right ) t}+c_{2} {\mathrm e}^{-t \sqrt {8 a +25}-t} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\left (-1+\sqrt {8 a +25}\right ) t} \sqrt {8 a +25}-c_{2} {\mathrm e}^{-t \sqrt {8 a +25}-t} \sqrt {8 a +25}-5 c_{1} {\mathrm e}^{\left (-1+\sqrt {8 a +25}\right ) t}-5 c_{2} {\mathrm e}^{-t \sqrt {8 a +25}-t}}{a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 188

DSolve[{D[x[t],t]==4*x[t]+a*y[t],D[y[t],t]==8*x[t]-6*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {e^{-\left (\left (\sqrt {8 a+25}+1\right ) t\right )} \left (c_1 \left (\left (\sqrt {8 a+25}+5\right ) e^{2 \sqrt {8 a+25} t}+\sqrt {8 a+25}-5\right )+a c_2 \left (e^{2 \sqrt {8 a+25} t}-1\right )\right )}{2 \sqrt {8 a+25}} \\ y(t)\to \frac {e^{-\left (\left (\sqrt {8 a+25}+1\right ) t\right )} \left (8 c_1 \left (e^{2 \sqrt {8 a+25} t}-1\right )+c_2 \left (\left (\sqrt {8 a+25}-5\right ) e^{2 \sqrt {8 a+25} t}+\sqrt {8 a+25}+5\right )\right )}{2 \sqrt {8 a+25}} \\ \end{align*}