76.8.16 problem 16

Internal problem ID [17505]
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 : 16
Date solved : Tuesday, January 28, 2025 at 10:39:39 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=\frac {5 x \left (t \right )}{4}+\frac {3 y \left (t \right )}{4}\\ \frac {d}{d t}y \left (t \right )&=a x \left (t \right )+\frac {5 y \left (t \right )}{4} \end{align*}

✓ Solution by Maple

Time used: 0.081 (sec). Leaf size: 82

dsolve([diff(x(t),t)=5/4*x(t)+3/4*y(t),diff(y(t),t)=a*x(t)+5/4*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (5+2 \sqrt {3}\, \sqrt {a}\right ) t}{4}}+c_{2} {\mathrm e}^{-\frac {\left (2 \sqrt {3}\, \sqrt {a}-5\right ) t}{4}} \\ y \left (t \right ) &= \frac {2 \sqrt {3}\, \sqrt {a}\, \left (c_{1} {\mathrm e}^{\frac {\left (5+2 \sqrt {3}\, \sqrt {a}\right ) t}{4}}-c_{2} {\mathrm e}^{-\frac {\left (2 \sqrt {3}\, \sqrt {a}-5\right ) t}{4}}\right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 164

DSolve[{D[x[t],t]==5/4*x[t]+3/4*y[t],D[y[t],t]==a*x[t]+5/4*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {e^{\frac {1}{4} \left (5-2 \sqrt {3} \sqrt {a}\right ) t} \left (2 \sqrt {a} c_1 \left (e^{\sqrt {3} \sqrt {a} t}+1\right )+\sqrt {3} c_2 \left (e^{\sqrt {3} \sqrt {a} t}-1\right )\right )}{4 \sqrt {a}} \\ y(t)\to \frac {1}{6} e^{\frac {1}{4} \left (5-2 \sqrt {3} \sqrt {a}\right ) t} \left (2 \sqrt {3} \sqrt {a} c_1 \left (e^{\sqrt {3} \sqrt {a} t}-1\right )+3 c_2 \left (e^{\sqrt {3} \sqrt {a} t}+1\right )\right ) \\ \end{align*}