64.1.10 problem 2.2 (iv)

Internal problem ID [15587]
Book : Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999. Oxford Univ. Press. NY
Section : Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number : 2.2 (iv)
Date solved : Thursday, October 02, 2025 at 10:20:28 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \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: 0.156 (sec). Leaf size: 23

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

\begin{align*} x \left (t \right ) &= c_2 \,{\mathrm e}^{t} \\ y \left (t \right ) &= -\frac {c_2 \,{\mathrm e}^{t}}{2}+c_1 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 39

ode={D[x[t],t]==x[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 c_1 e^t\\ y(t)&\to \left (\frac {c_1}{2}+c_2\right ) e^{3 t}-\frac {c_1 e^t}{2} \end{align*}

✓ Sympy. Time used: 0.035 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(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)
 

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