For the y=kx question, you could use the particular solutions for x and y and rearrange to get an equation of the form ae^-3t(bsint+ccost)=0, and since e^-3t>0 for all t, then bsint+ccost=0. This gives tanx=-c/b, and since tant is periodic, there are infinitely many solutions for t such that y=kx (I think that k was equal to 4, so y=4x for infinitely mant t). I remember getting something like tant=-14/5, although this may not be correct because I don't remember much of it!
Did anyone get y=-x^2(1+cosx) for the first part of Q3? I think it is correct after checking on Wolfram Alpha, but I thought that the graph was a bit of a pain! The graph looks like oscillations of increasing amplitude, all below the x-axis, touching the x-axis at x=0,pi,3pi.