52.9.13 problem 13

Internal problem ID [8391]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.1. Page 332
Problem number : 13
Date solved : Sunday, March 30, 2025 at 01:00:37 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.102 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 66

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

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

✓ Sympy. Time used: 0.108 (sec). Leaf size: 37

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

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