70.7.7 problem 7

Internal problem ID [18767]
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 3.3 (Homogeneous linear systems with constant coefficients). Problems at page 165
Problem number : 7
Date solved : Thursday, October 02, 2025 at 03:30:42 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 )&=\frac {3 x \left (t \right )}{4}+\frac {5 y \left (t \right )}{4} \end{align*}

✓ Maple. Time used: 0.115 (sec). Leaf size: 34

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {t}{2}}+c_2 \,{\mathrm e}^{2 t} \\ y \left (t \right ) &= -c_1 \,{\mathrm e}^{\frac {t}{2}}+c_2 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 80

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

\begin{align*} x(t)&\to \frac {1}{2} e^{t/2} \left (c_1 \left (e^{3 t/2}+1\right )+c_2 \left (e^{3 t/2}-1\right )\right )\\ y(t)&\to \frac {1}{2} e^{t/2} \left (c_1 \left (e^{3 t/2}-1\right )+c_2 \left (e^{3 t/2}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.052 (sec). Leaf size: 31

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

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