54.10.10 problem 1924

Internal problem ID [13144]
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, October 12, 2025 at 02:35:48 AM
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: 0.04 (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