75.26.4 problem 771
Internal
problem
ID
[17153]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Chapter
3
(Systems
of
differential
equations).
Section
19.
Basic
concepts
and
definitions.
Exercises
page
199
Problem
number
:
771
Date
solved
:
Friday, March 14, 2025 at 04:49:22 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=\frac {x_{1} \left (t \right )^{2}}{x_{2} \left (t \right )}\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{2} \left (t \right )-x_{1} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.112 (sec). Leaf size: 66
ode:=[diff(x__1(t),t) = x__1(t)^2/x__2(t), diff(x__2(t),t) = x__2(t)-x__1(t)];
dsolve(ode);
\begin{align*}
\\
\left [\left \{x_{1} \left (t \right ) &= \frac {1}{\sqrt {2 \,{\mathrm e}^{-t} c_{1} -2 c_{2}}}, x_{1} \left (t \right ) = -\frac {1}{\sqrt {2 \,{\mathrm e}^{-t} c_{1} -2 c_{2}}}\right \}, \left \{x_{2} \left (t \right ) = \frac {x_{1} \left (t \right )^{2}}{\frac {d}{d t}x_{1} \left (t \right )}\right \}\right ] \\
\end{align*}
✓ Mathematica. Time used: 0.099 (sec). Leaf size: 125
ode={D[ x1[t],t]==x1[t]^2/x2[t],D[ x2[t],t]==x2[t]-x1[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x2}(t)\to \frac {2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (2 c_1 K[1]^2+1\right )}dK[1]\&\right ]\left [\frac {t}{2}+c_2\right ]}{1+2 c_1 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (2 c_1 K[1]^2+1\right )}dK[1]\&\right ]\left [\frac {t}{2}+c_2\right ]{}^2} \\
\text {x1}(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (2 c_1 K[1]^2+1\right )}dK[1]\&\right ]\left [\frac {t}{2}+c_2\right ] \\
\end{align*}
✓ Sympy. Time used: 0.520 (sec). Leaf size: 112
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(-x__1(t)**2/x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) - x__2(t) + Derivative(x__2(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = \sqrt {\frac {e^{C_{2} + t}}{C_{1} \left (1 - e^{C_{2} + t}\right )}}, \ x^{2}{\left (t \right )} = \frac {2 \sqrt {\frac {e^{C_{2} + t}}{C_{1} \left (1 - e^{C_{2} + t}\right )}}}{1 + \frac {e^{C_{2} + t}}{1 - e^{C_{2} + t}}}, \ x^{1}{\left (t \right )} = - \sqrt {- \frac {e^{C_{2} + t}}{C_{1} \left (e^{C_{2} + t} - 1\right )}}, \ x^{2}{\left (t \right )} = - \frac {2 \sqrt {- \frac {e^{C_{2} + t}}{C_{1} \left (e^{C_{2} + t} - 1\right )}}}{1 - \frac {e^{C_{2} + t}}{e^{C_{2} + t} - 1}}\right ]
\]