56.1.75 problem 75

Internal problem ID [8787]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 75
Date solved : Wednesday, March 05, 2025 at 06:48:28 AM
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 )\\ \frac {d}{d t}z \left (t \right )&=z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 26

ode:=[diff(x(t),t) = x(t)+y(t), diff(y(t),t) = y(t), diff(z(t),t) = z(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} \\ z \left (t \right ) &= c_3 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 62

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

\begin{align*} x(t)\to e^t (c_2 t+c_1) \\ y(t)\to c_2 e^t \\ z(t)\to c_3 e^t \\ x(t)\to e^t (c_2 t+c_1) \\ y(t)\to c_2 e^t \\ z(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.077 (sec). Leaf size: 27

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

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