85.84.3 problem 2 (b)

Internal problem ID [23014]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. B Exercises at page 445
Problem number : 2 (b)
Date solved : Sunday, October 12, 2025 at 05:55:06 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 )^{2}+1\\ \frac {d}{d t}z \left (t \right )&=z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.078 (sec). Leaf size: 28

ode:=[diff(x(t),t) = x(t)*y(t), diff(y(t),t) = y(t)^2+1, diff(z(t),t) = z(t)]; 
dsolve(ode);
 

\begin{align*} \{z \left (t \right ) &= c_3 \,{\mathrm e}^{t}\} \\ \{y \left (t \right ) = \tan \left (t +c_2 \right )\} \\ \{x \left (t \right ) &= c_1 \,{\mathrm e}^{\int y \left (t \right )d t}\} \\ \end{align*}

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 54

ode={D[x[t],{t,1}]==x[t]*y[t],D[y[t],{t,1}]==y[t]^2+1,D[z[t],t]==z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

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

✗ Sympy

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)**2 + Derivative(y(t), t) - 1,0),Eq(-z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

KeyError : F2_