60.10.4 problem 1916
Internal
problem
ID
[11839]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
9,
system
of
higher
order
odes
Problem
number
:
1916
Date
solved
:
Friday, March 14, 2025 at 03:00:12 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=h \left (a -x \left (t \right )\right ) \left (c -x \left (t \right )-y \left (t \right )\right )\\ \frac {d}{d t}y \left (t \right )&=k \left (b -y \left (t \right )\right ) \left (c -x \left (t \right )-y \left (t \right )\right ) \end{align*}
✓ Maple. Time used: 1.425 (sec). Leaf size: 237
ode:=[diff(x(t),t) = h*(a-x(t))*(c-x(t)-y(t)), diff(y(t),t) = k*(b-y(t))*(c-x(t)-y(t))];
dsolve(ode);
\begin{align*}
\left [\{x \left (t \right ) = a\}, \left \{y \left (t \right ) &= -\frac {a \,{\mathrm e}^{a c_{1} k +a k t +b c_{1} k +b k t -c c_{1} k -c k t}-c \,{\mathrm e}^{a c_{1} k +a k t +b c_{1} k +b k t -c c_{1} k -c k t}+b}{-1+{\mathrm e}^{a c_{1} k +a k t +b c_{1} k +b k t -c c_{1} k -c k t}}\right \}\right ] \\
\left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a} -a \right )^{-\frac {k}{h}}}{\left (h \left (\textit {\_a} -a \right )^{-\frac {k}{h}} \textit {\_a} +h \left (\textit {\_a} -a \right )^{-\frac {k}{h}} b -h \left (\textit {\_a} -a \right )^{-\frac {k}{h}} c +c_{1} \right ) \left (\textit {\_a} -a \right )}d \textit {\_a} +t +c_{2} \right )\right \}, \left \{y \left (t \right ) = \frac {-x \left (t \right )^{2} h +x \left (t \right ) a h +x \left (t \right ) c h -a c h +\frac {d}{d t}x \left (t \right )}{h x \left (t \right )-a h}\right \}\right ] \\
\end{align*}
✓ Mathematica. Time used: 0.445 (sec). Leaf size: 277
ode={D[x[t],t]==h*(a-x[t])*(c-x[t]-y[t]),D[y[t],t]==k*(b-y[t])*(c-x[t]-y[t])};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
y(t)\to b+c_1 \left (h \left (a-\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {(h (a-K[1]))^{\frac {k}{h}}}{(a-K[1]) \left (c_1 (a h-h K[1])^{\frac {k}{h}} (h (a-K[1]))^{\frac {k}{h}}-c (h (a-K[1]))^{\frac {k}{h}}+K[1] (h (a-K[1]))^{\frac {k}{h}}+b (a h-h K[1])^{\frac {k}{h}}\right )}dK[1]\&\right ][-h t+c_2]\right )\right ){}^{\frac {k}{h}} \\
x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {(h (a-K[1]))^{\frac {k}{h}}}{(a-K[1]) \left (c_1 (a h-h K[1])^{\frac {k}{h}} (h (a-K[1]))^{\frac {k}{h}}-c (h (a-K[1]))^{\frac {k}{h}}+K[1] (h (a-K[1]))^{\frac {k}{h}}+b (a h-h K[1])^{\frac {k}{h}}\right )}dK[1]\&\right ][-h t+c_2] \\
\end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
a = symbols("a")
b = symbols("b")
c = symbols("c")
h = symbols("h")
k = symbols("k")
x = Function("x")
y = Function("y")
ode=[Eq(-h*(a - x(t))*(c - x(t) - y(t)) + Derivative(x(t), t),0),Eq(-k*(b - y(t))*(c - x(t) - y(t)) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)