60.10.14 problem 1928

Internal problem ID [11849]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1928
Date solved : Sunday, March 30, 2025 at 09:18:28 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )&=\frac {k x \left (t \right )}{\left (x \left (t \right )^{2}+y \left (t \right )^{2}\right )^{{3}/{2}}}\\ \frac {d^{2}}{d t^{2}}y \left (t \right )&=\frac {k y \left (t \right )}{\left (x \left (t \right )^{2}+y \left (t \right )^{2}\right )^{{3}/{2}}} \end{align*}

✗ Maple

ode:=[diff(diff(x(t),t),t) = k*x(t)/(x(t)^2+y(t)^2)^(3/2), diff(diff(y(t),t),t) = k*y(t)/(x(t)^2+y(t)^2)^(3/2)]; 
dsolve(ode);
 

\[ \text {No solution found} \]

✗ Mathematica

ode={D[x[t],{t,2}]==k*x[t]/(x[t]^2+y[t]^2)^(3/2),D[y[t],{t,2}]==k*y[t]/(x[t]^2+y[t]^2)^(3/2)}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
t = symbols("t") 
k = symbols("k") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-k*x(t)/(x(t)**2 + y(t)**2)**(3/2) + Derivative(x(t), (t, 2)),0),Eq(-k*y(t)/(x(t)**2 + y(t)**2)**(3/2) + Derivative(y(t), (t, 2)),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

NotImplementedError :