46.10.14 problem 13

Internal problem ID [9694]
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 8.2. Page 346
Problem number : 13
Date solved : Tuesday, September 30, 2025 at 06:29:06 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=4 \\ y \left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.146 (sec). Leaf size: 24

ode:=[diff(x(t),t) = 1/2*x(t), diff(y(t),t) = x(t)-1/2*y(t)]; 
ic:=[x(0) = 4, y(0) = 5]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 32

ode={D[x[t],t]==1/2*x[t],D[y[t],t]==x[t]-1/2*y[t]}; 
ic={x[0]==4,y[0]==5}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.048 (sec). Leaf size: 24

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

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