52.10.33 problem 36

Internal problem ID [8427]
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 : 36
Date solved : Sunday, March 30, 2025 at 01:04:36 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )+5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+6 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.140 (sec). Leaf size: 58

ode:=[diff(x(t),t) = 4*x(t)+5*y(t), diff(y(t),t) = -2*x(t)+6*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{5 t} \left (\sin \left (3 t \right ) c_1 +\cos \left (3 t \right ) c_2 \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{5 t} \left (\sin \left (3 t \right ) c_1 -3 \sin \left (3 t \right ) c_2 +3 \cos \left (3 t \right ) c_1 +\cos \left (3 t \right ) c_2 \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 69

ode={D[x[t],t]==4*x[t]+5*y[t],D[y[t],t]==-2*x[t]+6*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{5 t} (3 c_1 \cos (3 t)-(c_1-5 c_2) \sin (3 t)) \\ y(t)\to \frac {1}{3} e^{5 t} (3 c_2 \cos (3 t)+(c_2-2 c_1) \sin (3 t)) \\ \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) - 5*y(t) + Derivative(x(t), t),0),Eq(2*x(t) - 6*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {C_{1}}{2} - \frac {3 C_{2}}{2}\right ) e^{5 t} \sin {\left (3 t \right )} + \left (\frac {3 C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{5 t} \cos {\left (3 t \right )}, \ y{\left (t \right )} = - C_{1} e^{5 t} \sin {\left (3 t \right )} + C_{2} e^{5 t} \cos {\left (3 t \right )}\right ] \]