76.7.9 problem 9

Internal problem ID [17411]
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 : 9
Date solved : Monday, March 31, 2025 at 04:13:09 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-\frac {x \left (t \right )}{4}-\frac {3 y \left (t \right )}{4}\\ \frac {d}{d t}y \left (t \right )&=\frac {x \left (t \right )}{2}+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.113 (sec). Leaf size: 35

ode:=[diff(x(t),t) = -1/4*x(t)-3/4*y(t), diff(y(t),t) = 1/2*x(t)+y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {t}{2}}+c_2 \,{\mathrm e}^{\frac {t}{4}} \\ y \left (t \right ) &= -c_1 \,{\mathrm e}^{\frac {t}{2}}-\frac {2 c_2 \,{\mathrm e}^{\frac {t}{4}}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 81

ode={D[x[t],t]==-1/4*x[t]-3/4*y[t],D[y[t],t]==1/2*x[t]+y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.091 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t)/4 + 3*y(t)/4 + Derivative(x(t), t),0),Eq(-x(t)/2 - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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