70.8.16 problem 16

Internal problem ID [18792]
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 : Thursday, October 02, 2025 at 03:30:56 PM
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*}

✓ Maple. Time used: 0.124 (sec). Leaf size: 82

ode:=[diff(x(t),t) = 5/4*x(t)+3/4*y(t), diff(y(t),t) = a*x(t)+5/4*y(t)]; 
dsolve(ode);
 

\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*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 164

ode={D[x[t],t]==5/4*x[t]+3/4*y[t],D[y[t],t]==a*x[t]+5/4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.124 (sec). Leaf size: 109

from sympy import * 
t = symbols("t") 
a = symbols("a") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t)/4 - 3*y(t)/4 + Derivative(x(t), t),0),Eq(-a*x(t) - 5*y(t)/4 + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {\sqrt {3} C_{1} e^{- \frac {t \left (2 \sqrt {3} \sqrt {a} - 5\right )}{4}}}{2 \sqrt {a}} + \frac {\sqrt {3} C_{2} e^{\frac {t \left (2 \sqrt {3} \sqrt {a} + 5\right )}{4}}}{2 \sqrt {a}}, \ y{\left (t \right )} = C_{1} e^{- \frac {t \left (2 \sqrt {3} \sqrt {a} - 5\right )}{4}} + C_{2} e^{\frac {t \left (2 \sqrt {3} \sqrt {a} + 5\right )}{4}}\right ] \]