A particle moves in a plane under a force, towards a fixed centre, proportional to the distance. If the path of the particle has two apsidal distances $a,b$ $(a>b)$, then find the equation of the path.
I came across this question on MSE. While solving I could only proceed to the step
$$ h^2(du/dθ^2+u^2)=(k/u^2)+C $$
where $C$ is constant of integration and the force is considered as
$$\mathbf F=-k\mathbf r.$$
I also know that at an apse $du/dθ=0$. How do I proceed further?
EDIT: I have found out the pedal equation of the orbit and it seems to be an ellipse. How do I find out in terms of polar coordinates $u(=1/r),θ$? It is required to prove that $$ u^2=(sin^2θ)/b^2+(cos^2θ)/a^2 $$ Here's a link to the original question Orbits under central forces
The system $$ m\ddot{\mathbf r} = -k\mathbf r $$ that you seem to describe consists of two decoupled harmonic oscillators. It has solutions $x(t)=A\sin(\omega t+\phi_1)$, $y(t)=B\sin(ωt+ϕ_2)$, $ω^2=\frac{k}m$.
If you set $A=a$, $B=b$, $ϕ_1=0$ and $ϕ_2=\frac\pi2$, then you get a horizontal ellipse with main axes with half-lengths $a$ and $b$.