having trouble finding the explicit function for a pursuit problem

Question

I was trying to solve the following problem.

A mouse is located at the origin of the coordinate plane and a cat is directly 10 units below it. Then the mouse starts traveling at a constant speed of 6 along the positive x-axis, and the cat starts traveling at a constant speed of 8 but the direction of the cat is always directed towards the mouse at any instant. Find the explicit equation that traces out the path traveled by the cat at each point in time and also find the equation of the path traveled by the cat without the parameter time(i.e. only in terms of x and y).

I tried to do this rather directly as possible. let $\overrightarrow{R}(t)$ be the position of the mouse at each point in time. Let $\overrightarrow{V_c}(t)$ be the velocity of the cat at each point in time and let $\overrightarrow{C}(t)$ be the position of the cat at each point in time. Another piece of information we know is that $||\overrightarrow{V_c}(t)||$ is 8 for all t and $\overrightarrow{C}(0)=-10\hat{j}$.

So I set up the necessary differential equation.$$\frac{d}{dt}\overrightarrow{C}(t)=\frac{8[\overrightarrow{R}(t)-C(t)]}{||\overrightarrow{R}(t)-C(t)||}$$expanding it out you get:$$\frac{d}{dt}[x_c(t)\hat{i}+y_c(t)\hat{j}]=\frac{8[6t-x_c(t)]\hat{i}+8[-y_c(t)]\hat{j}}{\sqrt{(6t-x_c(t))^2+(y_c(t))^2}}$$After this point I am completely clueless what to do how do you solve this differential equation? Is there any known analytic solution to this differential equation or would you solve it numerically Help!!!???