65.15.3 problem 28.2 (iii)

Internal problem ID [13756]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 28, Distinct real eigenvalues. Exercises page 282
Problem number : 28.2 (iii)
Date solved : Wednesday, March 05, 2025 at 10:14:59 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=11 x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )+4 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 35

ode:=[diff(x(t),t) = 11*x(t)-2*y(t), diff(y(t),t) = 3*x(t)+4*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{10 t}+c_{2} {\mathrm e}^{5 t} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{10 t}}{2}+3 c_{2} {\mathrm e}^{5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 95

ode={D[x[t],t]==2*x[t]-2*y[t],D[y[t],t]==3*x[t]+4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.089 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-11*x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-3*x(t) - 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} e^{5 t}}{3} + 2 C_{2} e^{10 t}, \ y{\left (t \right )} = C_{1} e^{5 t} + C_{2} e^{10 t}\right ] \]