FP3 DEs by substitution question

Can anyone help me with this question? I can get as far as du/dx = u^3/(3x)-4u/(3x) but I'm not sure if it's correct up to here and if it is I'm struggling to integrate the separated variables.

Part D:

Could you post your working?

Just checking the end - can you collect the ln with the RHS?

You are making hard work of the partial fractions - why did you expand?

1 = A (u + 2)(u -2) + Bu (u - 2) + Cu (u + 2)

then subst in u = 0 then other sensible values - much quicker.

Anyway, they got the same as me in the back apart from a few questions I have.

I've integrated 1/3x incorrectly, but it's kind of confusing me. 1/3 could be taken out of the integrand and then it integrates to 1/3ln(X), however if the 3 is left in the integrand it integrates to 1/3ln(3x). Does the fraction have to be in its simplest terms to integrate like this idk.

Also, the attached answers aren't in the form y=f(X) which I thought was necessary. At what point is it ok to stop and not put it in that form?

What happens if you differentiate 1/3 ln(3x) .... you don't get 1/3x, so its best to take out the 1/3.

Once you subsitute back it usually easy to see if the answer can be given as y = f(x) and here it can't. So just tidy up as they have done.

ok awesome thank you!

Can I be cheeky and ask you what I've done wrong here too?? (what looks like 1/(8p) is meant to be (1/8)p btw

Of course!

Again, it's a slip I think - if you integrate 2dp you get 2p + c so if you integrate (1/8)dp you get (1/8)p not 1/8p.

I always suggest differentiating if you aren't sure or want to check.