76.8.18 problem 18

Internal problem ID [17507]
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 : 18
Date solved : Tuesday, January 28, 2025 at 10:39:41 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )+a y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-6 x \left (t \right )-4 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.099 (sec). Leaf size: 121

dsolve([diff(x(t),t)=3*x(t)+a*y(t),diff(y(t),t)=-6*x(t)-4*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {49-24 a}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {49-24 a}\right ) t}{2}} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {49-24 a}\right ) t}{2}} \sqrt {49-24 a}-c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {49-24 a}\right ) t}{2}} \sqrt {49-24 a}-7 c_{1} {\mathrm e}^{\frac {\left (-1+\sqrt {49-24 a}\right ) t}{2}}-7 c_{2} {\mathrm e}^{-\frac {\left (1+\sqrt {49-24 a}\right ) t}{2}}}{2 a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 189

DSolve[{D[x[t],t]==3*x[t]+a*y[t],D[y[t],t]==-6*x[t]-4*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {49-24 a}+1\right ) t} \left (c_1 \left (\left (\sqrt {49-24 a}+7\right ) e^{\sqrt {49-24 a} t}+\sqrt {49-24 a}-7\right )+2 a c_2 \left (e^{\sqrt {49-24 a} t}-1\right )\right )}{2 \sqrt {49-24 a}} \\ y(t)\to \frac {e^{-\frac {1}{2} \left (\sqrt {49-24 a}+1\right ) t} \left (c_2 \left (\left (\sqrt {49-24 a}-7\right ) e^{\sqrt {49-24 a} t}+\sqrt {49-24 a}+7\right )-12 c_1 \left (e^{\sqrt {49-24 a} t}-1\right )\right )}{2 \sqrt {49-24 a}} \\ \end{align*}