The radial and transverse equations of motion are (H is a constant). Solving these is not easy. I'm guessing that you're supposed to know enough about orbital motion to know that these equations imply that , where s=1/r. (You can find a derivation of that equation here—see Section 2 on that page.)
You are given that f(r) = kr, so the equation becomes . I don't know how to solve that differential equation. But looking at part (a) of the question, you can see that the solution is supposed to be of the form . -----(*)
So it makes sense to introduce a new variable . Then , with derivatives and (where dashes indicate differentiation with respect to theta). The differential equation can then be written in terms of u as . I don't know of a constructive way to solve that differential equation either. But by substituting the formula (*) for u, you can verify that it does indeed satisfy the equation, subject to some condition like . -----(**)
You haven't said what the constants E and L are. But the equation (**) above, together with the initial conditions of the problem, ought to give the answers here.
The equation can be written , which you can put into Cartesian form in the usual way ( , , ).
Intuition suggests a hyperbola, since the force is repulsive.