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STEP Support Programme

S2 2014 Q3

Solving the differential equation in part ii, I'm unsure what happens to the constant term - or why there isnt one?
Also in the solutions what does "straight lines... in the direction of the tangents to the circle at that point" mean? Would the lines meet?

You do have the arbitrary constant term. In the case where $y'' = 0$ you get the solution $y=mx+c$ for two arbitrary constants $m,c$. But you have some extra information, so you can determine $c$. In particular simply plug $y$ back into $(\ast)$ to get $c$ in terms of $m$.

In the other case, you end up with $y^2 = k(x^2-a^2)$ (or something similar) for some arbitrary constant $k$ which yoan then show is infact $-1$ since the distance to the origin is $a^2$.

The lines would necessarily have to meet at some point. But they mean that you need to have two tangent lines on either side of the circle. It's hard to convey this without a drawing - let me know if you're still confused and I'll see what I can do.

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