14.27.3 problem 3

Internal problem ID [2791]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.1 (Introduction). Page 3770
Problem number : 3
Date solved : Sunday, March 30, 2025 at 12:20:34 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=a x \left (t \right )-b x \left (t \right ) y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-c y \left (t \right )+d x \left (t \right ) y \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=z \left (t \right )+x \left (t \right )^{2}+y \left (t \right )^{2} \end{align*}

✓ Maple. Time used: 0.676 (sec). Leaf size: 171

ode:=[diff(x(t),t) = a*x(t)-b*x(t)*y(t), diff(y(t),t) = -c*y(t)+d*x(t)*y(t), diff(z(t),t) = z(t)+x(t)^2+y(t)^2]; 
dsolve(ode);
 

\begin{align*} \\ \left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} a \left (\operatorname {LambertW}\left (\frac {{\mathrm e}^{-1} \textit {\_a}^{-\frac {c}{a}} {\mathrm e}^{\frac {\textit {\_a} d}{a}} {\mathrm e}^{\frac {c_2}{a}}}{a}\right )+1\right )}d \textit {\_a} +t +c_3 \right )\right \}, \left \{y \left (t \right ) = \frac {a x \left (t \right )-\frac {d}{d t}x \left (t \right )}{x \left (t \right ) b}\right \}, \left \{z \left (t \right ) = \left (\int \frac {{\mathrm e}^{-t} \left (x \left (t \right )^{4} b^{2}+a^{2} x \left (t \right )^{2}-2 \left (\frac {d}{d t}x \left (t \right )\right ) a x \left (t \right )+\left (\frac {d}{d t}x \left (t \right )\right )^{2}\right )}{x \left (t \right )^{2} b^{2}}d t +c_1 \right ) {\mathrm e}^{t}\right \}\right ] \\ \end{align*}

✓ Mathematica. Time used: 0.352 (sec). Leaf size: 459

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

\begin{align*} y(t)\to -\frac {a W\left (-\frac {b \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][a t+c_2]{}^{-\frac {c}{a}} \exp \left (\frac {d \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][a t+c_2]-c_1}{a}\right )}{a}\right )}{b} \\ x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][a t+c_2] \\ z(t)\to e^t \left (\int _1^t\frac {e^{-K[2]} \left (a^2 W\left (-\frac {b \exp \left (\frac {d \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][c_2+a K[2]]-c_1}{a}\right ) \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][c_2+a K[2]]{}^{-\frac {c}{a}}}{a}\right ){}^2+b^2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][c_2+a K[2]]{}^2\right )}{b^2}dK[2]+c_3\right ) \\ \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-a*x(t) + b*x(t)*y(t) + Derivative(x(t), t),0),Eq(c*y(t) - d*x(t)*y(t) + Derivative(y(t), t),0),Eq(-x(t)**2 - y(t)**2 - z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

NotImplementedError :