70.7.8 problem 8

Internal problem ID [18768]
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 : 8
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 {3 x \left (t \right )}{4}-\frac {7 y \left (t \right )}{4}\\ \frac {d}{d t}y \left (t \right )&=\frac {x \left (t \right )}{4}+\frac {5 y \left (t \right )}{4} \end{align*}

✓ Maple. Time used: 0.106 (sec). Leaf size: 31

ode:=[diff(x(t),t) = -3/4*x(t)-7/4*y(t), diff(y(t),t) = 1/4*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}^{t} \\ y \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{-\frac {t}{2}}}{7}-c_2 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 84

ode={D[x[t],t]==-3/4*x[t]-7/4*y[t],D[y[t],t]==1/4*x[t]+5/4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{6} e^{-t/2} \left (-\left (c_1 \left (e^{3 t/2}-7\right )\right )-7 c_2 \left (e^{3 t/2}-1\right )\right )\\ y(t)&\to \frac {1}{6} e^{-t/2} \left (c_1 \left (e^{3 t/2}-1\right )+c_2 \left (7 e^{3 t/2}-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.054 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t)/4 + 7*y(t)/4 + Derivative(x(t), t),0),Eq(-x(t)/4 - 5*y(t)/4 + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 7 C_{1} e^{- \frac {t}{2}} - C_{2} e^{t}, \ y{\left (t \right )} = C_{1} e^{- \frac {t}{2}} + C_{2} e^{t}\right ] \]