76.8.15 problem 15

Internal problem ID [17504]
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 : 15
Date solved : Tuesday, January 28, 2025 at 10:39:38 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=a x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.092 (sec). Leaf size: 99

dsolve([diff(x(t),t)=2*x(t)-5*y(t),diff(y(t),t)=a*x(t)-2*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} \sin \left (\sqrt {5 a -4}\, t \right )+c_{2} \cos \left (\sqrt {5 a -4}\, t \right ) \\ y \left (t \right ) &= -\frac {c_{1} \sqrt {5 a -4}\, \cos \left (\sqrt {5 a -4}\, t \right )}{5}+\frac {c_{2} \sqrt {5 a -4}\, \sin \left (\sqrt {5 a -4}\, t \right )}{5}+\frac {2 c_{1} \sin \left (\sqrt {5 a -4}\, t \right )}{5}+\frac {2 c_{2} \cos \left (\sqrt {5 a -4}\, t \right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 184

DSolve[{D[x[t],t]==2*x[t]-5*y[t],D[y[t],t]==a*x[t]-2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {e^{-\sqrt {4-5 a} t} \left (c_1 \left (\left (\sqrt {4-5 a}+2\right ) e^{2 \sqrt {4-5 a} t}+\sqrt {4-5 a}-2\right )-5 c_2 \left (e^{2 \sqrt {4-5 a} t}-1\right )\right )}{2 \sqrt {4-5 a}} \\ y(t)\to \frac {e^{-\sqrt {4-5 a} t} \left (a c_1 \left (e^{2 \sqrt {4-5 a} t}-1\right )+c_2 \left (\left (\sqrt {4-5 a}-2\right ) e^{2 \sqrt {4-5 a} t}+\sqrt {4-5 a}+2\right )\right )}{2 \sqrt {4-5 a}} \\ \end{align*}