52.10.32 problem 35

Internal problem ID [8426]
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 : 35
Date solved : Wednesday, March 05, 2025 at 05:47:34 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )+y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 46

ode:=[diff(x(t),t) = 5*x(t)+y(t), diff(y(t),t) = -2*x(t)+3*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{4 t} \left (c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )\right ) \\ y &= {\mathrm e}^{4 t} \left (\cos \left (t \right ) c_{1} -c_{2} \sin \left (t \right )-c_{2} \cos \left (t \right )-c_{1} \sin \left (t \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 51

ode={D[x[t],t]==5*x[t]+y[t],D[y[t],t]==-2*x[t]+3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.117 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) - y(t) + Derivative(x(t), t),0),Eq(2*x(t) - 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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