60.10.12 problem 1924
Internal
problem
ID
[11923]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
9,
system
of
higher
order
odes
Problem
number
:
1924
Date
solved
:
Tuesday, January 28, 2025 at 06:24:08 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 x \left (t \right ) t\\ \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 y \left (t \right ) t \end{align*}
✓ Solution by Maple
Time used: 1.454 (sec). Leaf size: 186
dsolve([(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)],singsol=all)
\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 (\left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )^{2}-t^{2} \left (\frac {d}{d t}x \left (t \right )\right )+2 x \left (t \right ) t \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 (\left (\frac {d}{d t}x \left (t \right )\right ) x \left (t \right )^{2}-t^{2} \left (\frac {d}{d t}x \left (t \right )\right )+2 x \left (t \right ) t \right )}}{\frac {d}{d t}x \left (t \right )}\right \}\right ] \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.064 (sec). Leaf size: 70
DSolve[{(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]},{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*}