85.84.2 problem 2 (a)

Internal problem ID [23013]
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 (a)
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 )&=y \left (t \right ) z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right ) z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right ) y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.401 (sec). Leaf size: 323

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

\begin{align*} [\{x \left (t \right ) = 0\}, \{y \left (t \right ) = 0\}, \{z \left (t \right ) = c_1\}] \\ [\{x \left (t \right ) = 0\}, \{y \left (t \right ) = c_1\}, \{z \left (t \right ) = 0\}] \\ [\{x \left (t \right ) = c_1\}, \{y \left (t \right ) = 0\}, \{z \left (t \right ) = 0\}] \\ \left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}-\frac {2}{\sqrt {4 \textit {\_a}^{4}-16 c_2 \,\textit {\_a}^{2}+16 c_2^{2}+c_1}}d \textit {\_a} +t +c_3 \right ), x \left (t \right ) = \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {2}{\sqrt {4 \textit {\_a}^{4}-16 c_2 \,\textit {\_a}^{2}+16 c_2^{2}+c_1}}d \textit {\_a} +t +c_3 \right )\right \}, \left \{y \left (t \right ) = -\frac {\sqrt {-2 x \left (t \right ) \left (-\frac {d^{2}}{d t^{2}}x \left (t \right )+\sqrt {\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2}-4 x \left (t \right )^{2} \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}, y \left (t \right ) = \frac {\sqrt {-2 x \left (t \right ) \left (-\frac {d^{2}}{d t^{2}}x \left (t \right )+\sqrt {\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2}-4 x \left (t \right )^{2} \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}, y \left (t \right ) = -\frac {\sqrt {2}\, \sqrt {x \left (t \right ) \left (\frac {d^{2}}{d t^{2}}x \left (t \right )+\sqrt {\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2}-4 x \left (t \right )^{2} \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}, y \left (t \right ) = \frac {\sqrt {2}\, \sqrt {x \left (t \right ) \left (\frac {d^{2}}{d t^{2}}x \left (t \right )+\sqrt {\left (\frac {d^{2}}{d t^{2}}x \left (t \right )\right )^{2}-4 x \left (t \right )^{2} \left (\frac {d}{d t}x \left (t \right )\right )^{2}}\right )}}{2 x \left (t \right )}\right \}, \left \{z \left (t \right ) = \frac {\frac {d}{d t}x \left (t \right )}{y \left (t \right )}\right \}\right ] \\ \end{align*}

✓ Mathematica. Time used: 0.196 (sec). Leaf size: 601

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

\begin{align*} x(t)&\to -i \sqrt {2} \sqrt {c_1} \text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right )\\ y(t)&\to -\sqrt {2} \sqrt {-c_1 \left (-1+\text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right ){}^2\right )}\\ z(t)&\to -\sqrt {2} \sqrt {c_2-c_1 \text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right ){}^2}\\ x(t)&\to -i \sqrt {2} \sqrt {c_1} \text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right )\\ y(t)&\to \sqrt {2} \sqrt {-c_1 \left (-1+\text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right ){}^2\right )}\\ z(t)&\to \sqrt {2} \sqrt {c_2-c_1 \text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right ){}^2}\\ x(t)&\to i \sqrt {2} \sqrt {c_1} \text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right )\\ y(t)&\to -\sqrt {2} \sqrt {-c_1 \left (-1+\text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right ){}^2\right )}\\ z(t)&\to \sqrt {2} \sqrt {c_2-c_1 \text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right ){}^2}\\ x(t)&\to i \sqrt {2} \sqrt {c_1} \text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right )\\ y(t)&\to \sqrt {2} \sqrt {-c_1 \left (-1+\text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right ){}^2\right )}\\ z(t)&\to -\sqrt {2} \sqrt {c_2-c_1 \text {sn}\left (i \sqrt {2} \sqrt {c_2} (t-c_3)|\frac {c_1}{c_2}\right ){}^2} \end{align*}

✗ Sympy

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

KeyError : F2_