50.27.5 problem 5

Internal problem ID [8186]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section 10.2 Linear Systems. Page 380
Problem number : 5
Date solved : Sunday, March 30, 2025 at 12:47:54 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.145 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 25

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

\begin{align*} x(t)\to e^t (c_2 t+c_1) \\ y(t)\to c_2 e^t \\ \end{align*}

✓ Sympy. Time used: 0.071 (sec). Leaf size: 20

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

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