52.9.2 problem 2

Internal problem ID [8380]
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 8.1. Page 332
Problem number : 2
Date solved : Sunday, March 30, 2025 at 12:53:27 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )-7 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=5 x \left (t \right ) \end{align*}

✓ Maple. Time used: 0.148 (sec). Leaf size: 77

ode:=[diff(x(t),t) = 4*x(t)-7*y(t), diff(y(t),t) = 5*x(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (\sin \left (\sqrt {31}\, t \right ) \sqrt {31}\, c_2 -\cos \left (\sqrt {31}\, t \right ) \sqrt {31}\, c_1 -2 \sin \left (\sqrt {31}\, t \right ) c_1 -2 \cos \left (\sqrt {31}\, t \right ) c_2 \right )}{5} \\ y \left (t \right ) &= {\mathrm e}^{2 t} \left (\sin \left (\sqrt {31}\, t \right ) c_1 +\cos \left (\sqrt {31}\, t \right ) c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 98

ode={D[x[t],t]==4*x[t]-7*y[t],D[y[t],t]==5*x[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to c_1 e^{2 t} \cos \left (\sqrt {31} t\right )+\frac {(2 c_1-7 c_2) e^{2 t} \sin \left (\sqrt {31} t\right )}{\sqrt {31}} \\ y(t)\to c_2 e^{2 t} \cos \left (\sqrt {31} t\right )+\frac {(5 c_1-2 c_2) e^{2 t} \sin \left (\sqrt {31} t\right )}{\sqrt {31}} \\ \end{align*}

✓ Sympy. Time used: 0.204 (sec). Leaf size: 88

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) + 7*y(t) + Derivative(x(t), t),0),Eq(-5*x(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \left (\frac {2 C_{1}}{5} - \frac {\sqrt {31} C_{2}}{5}\right ) e^{2 t} \cos {\left (\sqrt {31} t \right )} - \left (\frac {\sqrt {31} C_{1}}{5} + \frac {2 C_{2}}{5}\right ) e^{2 t} \sin {\left (\sqrt {31} t \right )}, \ y{\left (t \right )} = C_{1} e^{2 t} \cos {\left (\sqrt {31} t \right )} - C_{2} e^{2 t} \sin {\left (\sqrt {31} t \right )}\right ] \]