52.10.35 problem 38

Internal problem ID [8429]
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 : 38
Date solved : Sunday, March 30, 2025 at 01:04:39 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-8 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.127 (sec). Leaf size: 57

ode:=[diff(x(t),t) = x(t)-8*y(t), diff(y(t),t) = x(t)-3*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 64

ode={D[x[t],t]==x[t]-8*y[t],D[y[t],t]==x[t]-3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-t} (c_1 \cos (2 t)+(c_1-4 c_2) \sin (2 t)) \\ y(t)\to \frac {1}{2} e^{-t} (2 c_2 \cos (2 t)+(c_1-2 c_2) \sin (2 t)) \\ \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 8*y(t) + Derivative(x(t), t),0),Eq(-x(t) + 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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