# 2014 STEP 2 Q3ii (Spoilers)

After differentiating * in (3ii),

I obtain y''=0 and

$\ a^2 y' + x(y-xy')\$ = 0

the above is a differential equation

$\ y' = \frac {xy}{x^2 - a^2}\$

I get the solution as

$\ y^2 = A (x^2 - a^2) \$

I have tried substituting the above into * given in question but this has not helped to figure out the constant

How do I show my solution for the differential equation is x^2 + y^2 = a^2 - as given in question ?

### Have you considered looking

Have you considered looking back at the property given in the first part of 3(ii): i.e $C$ has the property that the distance $a$ is the same for all points $P$ on $C$?

### Isn't this property

Isn't this property equivalent to the * equation - since I used this property to actually derive * ?

### Hi

Hi,

I like your method so far. I presume you got the equation of the straight line by substituting back into (*). It is possible (I'm not claiming it is the most efficient way) to find $A$ by substituting back into (*) also.

Every curve satisfying (*) must satisfy the resulting differential equations, but not every curve satisfying the resulting differential equations must satisfy (*), essentially because when we differentiate (*) we lose information about $y$. So when you substitute the solution back into (*) you are able to find which values of the constant are valid.

To make substituting back into (*) easier perhaps you might like to try differentiating $y^2 = A (x^2-a^2)$ implicitly first (i.e. $y' = Ax/y$) and then substitute since it avoids square roots (i.e. $y' = Ax/y$).

### Thank you; Substituting back

Thank you; Substituting back in * using your method, I get A=-1 as required

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