60.10.10 problem 1924
Internal
problem
ID
[11845]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
9,
system
of
higher
order
odes
Problem
number
:
1924
Date
solved
:
Sunday, March 30, 2025 at 09:18:24 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \left (x \left (t \right )^{2}+y \left (t \right )^{2}-t^{2}\right ) \left (\frac {d}{d t}x \left (t \right )\right )&=-2 t x \left (t \right )\\ \left (x \left (t \right )^{2}+y \left (t \right )^{2}-t^{2}\right ) \left (\frac {d}{d t}y \left (t \right )\right )&=-2 t y \left (t \right ) \end{align*}
✓ Maple. Time used: 0.647 (sec). Leaf size: 186
ode:=[(x(t)^2+y(t)^2-t^2)*diff(x(t),t) = -2*t*x(t), (x(t)^2+y(t)^2-t^2)*diff(y(t),t) = -2*t*y(t)];
dsolve(ode);
\begin{align*}
\left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) &= \frac {1+\sqrt {-4 c_1^{2} t^{2}+1}}{2 c_1}, y \left (t \right ) = -\frac {-1+\sqrt {-4 c_1^{2} t^{2}+1}}{2 c_1}\right \}\right ] \\
\left [\left \{x \left (t \right ) &= -\frac {-c_1 +\sqrt {-2 c_2 \,t^{2}+c_1^{2}}}{2 c_2}, x \left (t \right ) = \frac {c_1 +\sqrt {-2 c_2 \,t^{2}+c_1^{2}}}{2 c_2}\right \}, \left \{y \left (t \right ) = \frac {\sqrt {-\left (\frac {d}{d t}x \left (t \right )\right ) \left (x \left (t \right )^{2} \left (\frac {d}{d t}x \left (t \right )\right )-\left (\frac {d}{d t}x \left (t \right )\right ) t^{2}+2 t x \left (t \right )\right )}}{\frac {d}{d t}x \left (t \right )}, y \left (t \right ) = -\frac {\sqrt {-\left (\frac {d}{d t}x \left (t \right )\right ) \left (x \left (t \right )^{2} \left (\frac {d}{d t}x \left (t \right )\right )-\left (\frac {d}{d t}x \left (t \right )\right ) t^{2}+2 t x \left (t \right )\right )}}{\frac {d}{d t}x \left (t \right )}\right \}\right ] \\
\end{align*}
✓ Mathematica. Time used: 0.064 (sec). Leaf size: 70
ode={(x[t]^2+y[t]^2-t^2)*D[x[t],t]==-2*t*x[t],(x[t]^2+y[t]^2-t^2)*D[y[t],t]==-2*t*y[t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
y(t)\to c_1 x(t) \\
\text {Solve}\left [\int _1^{\frac {x(t)}{t}}\frac {c_1{}^2 K[1]^2+K[1]^2-1}{K[1] \left (c_1{}^2 K[1]^2+K[1]^2+1\right )}dK[1]&=-\log (t)+c_2,x(t)\right ] \\
\end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(2*t*x(t) + (-t**2 + x(t)**2 + y(t)**2)*Derivative(x(t), t),0),Eq(2*t*y(t) + (-t**2 + x(t)**2 + y(t)**2)*Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
TypeError : NoneType object is not subscriptable