52.10.5 problem 5

Internal problem ID [8399]
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 : 5
Date solved : Sunday, March 30, 2025 at 01:00:48 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=10 x \left (t \right )-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=8 x \left (t \right )-12 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.118 (sec). Leaf size: 35

ode:=[diff(x(t),t) = 10*x(t)-5*y(t), diff(y(t),t) = 8*x(t)-12*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{8 t}+c_2 \,{\mathrm e}^{-10 t} \\ y \left (t \right ) &= \frac {2 c_1 \,{\mathrm e}^{8 t}}{5}+4 c_2 \,{\mathrm e}^{-10 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 73

ode={D[x[t],t]==10*x[t]-5*y[t],D[y[t],t]==8*x[t]-12*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{18} e^{-10 t} \left (c_1 \left (20 e^{18 t}-2\right )-5 c_2 \left (e^{18 t}-1\right )\right ) \\ y(t)\to \frac {1}{9} e^{-10 t} \left (4 c_1 \left (e^{18 t}-1\right )-c_2 \left (e^{18 t}-10\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.093 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-10*x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(-8*x(t) + 12*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} e^{- 10 t}}{4} + \frac {5 C_{2} e^{8 t}}{2}, \ y{\left (t \right )} = C_{1} e^{- 10 t} + C_{2} e^{8 t}\right ] \]