Convert the following equations to polar coordinates (Newton's equations for a particle moving in a plane under the influence of a coulobic potential). I show what I did below, I am looking for confirmation or for correction.
$\tag{1} m \frac{d^2x}{dt^2}= \frac{-kx}{(x^2+y^2)^{3/2}}$
$\tag{2} m \frac{d^2y}{dt^2}= \frac{-ky}{(x^2+y^2)^{3/2}}$
NOTE: $x=r\cos\theta$, $y=r\sin\theta$, $r=(x^2+y^2)^{1/2}$, $\theta = \tan^-1(y/x)$
$\tag{3} m \frac{d^2x}{dt^2}= \frac{-kx}{(x^2+y^2)^{3/2}}$ $\tag{4} =\frac{-kr\cos\theta}{r^3}$ $\tag{5} = \frac{-k\cos\theta}{r^2}$
and similarly, $\tag{6} m \frac{d^2y}{dt^2} = \frac{-k\sin\theta}{r^2}$