60.10.12 problem 1926
Internal
problem
ID
[11847]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
9,
system
of
higher
order
odes
Problem
number
:
1926
Date
solved
:
Friday, March 14, 2025 at 03:00:15 AM
CAS
classification
:
system_of_ODEs
\begin{align*} x \left (t \right )&=t \left (\frac {d}{d t}x \left (t \right )\right )+f \left (\frac {d}{d t}x \left (t \right ), \frac {d}{d t}y \left (t \right )\right )\\ y \left (t \right )&=t \left (\frac {d}{d t}y \left (t \right )\right )+g \left (\frac {d}{d t}x \left (t \right ), \frac {d}{d t}y \left (t \right )\right ) \end{align*}
✓ Maple. Time used: 0.519 (sec). Leaf size: 95
ode:=[x(t) = t*diff(x(t),t)+f(diff(x(t),t),diff(y(t),t)), y(t) = t*diff(y(t),t)+g(diff(x(t),t),diff(y(t),t))];
dsolve(ode);
\begin{align*}
\{\int \operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right )d t +c_{1} &= \operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right ) t +f \left (\operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right ), \frac {d}{d t}y \left (t \right )\right )\} \\
\{x \left (t \right ) &= \int \operatorname {RootOf}\left (g \left (\textit {\_Z} , \frac {d}{d t}y \left (t \right )\right )-y \left (t \right )+t \left (\frac {d}{d t}y \left (t \right )\right )\right )d t +c_{1}\} \\
\end{align*}
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 28
ode={x[t]==t*D[x[t],t]+f[D[x[t],t],D[y[t],t]],y[t]==t*D[y[t],t]+g[D[x[t],t],D[y[t],t]]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to f(c_1,c_2)+c_1 t \\
y(t)\to g(c_1,c_2)+c_2 t \\
\end{align*}
✓ Sympy. Time used: 0.132 (sec). Leaf size: 17
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(-t*Derivative(x(t), t) - f(Derivative(x(t), t), Derivative(y(t), t)) + x(t),0),Eq(-t*Derivative(y(t), t) - g(Derivative(x(t), t), Derivative(y(t), t)) + y(t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left \{x{\left (t \right )} = C_{1} t + f{\left (C_{1},C_{2} \right )}, y{\left (t \right )} = C_{2} t + g{\left (C_{1},C_{2} \right )}\right \}
\]