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# Quasi-homogeneous 1st Order ODEs

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1. Wondering if anyone could explain this example to me, please?

I'm looking at a worked example in our notes and the qu is:

Solve dy/dx = (x + y + 3)/(x − y − 5).

The solution starts:

Put x = x0 + X, y = y0 + Y
We require x0 + y0 + 3 = 0 and x0 − y0 − 5 = 0 which has solution x0 = 1 and y0 = −4.
Then we have dY/dX = (X + Y)/(X − Y).

I understand all of that.

Then, it says:

Put Y = uX to get X*du/dX + u = (1 + u)/(1 − u).

I understand where the RHS came from... but how did they get the LHS of that?

And what happens to the 1 and -4 we found earlier?

Any help would be much appreciated. Thank you
2. differentiated uX with respect to X? (just a guess, i shouldnt really be looking at undergraduate)
3. (Original post by funkiichiicka)
I understand where the RHS came from... but how did they get the LHS of that?
The LHS is the product rule for differentiation. It is necessary to find .

(Original post by funkiichiicka)
What happened to the 1 and -4 we found earlier?
It was used to give us .

I hope that helps.

Darren
4. Yes, Y = uX, so .
5. (Original post by DPLSK)
The LHS is the product rule for differentiation. It is necessary to find .

It was used to give us .

I hope that helps.

Darren

Thank you.

I thought it was something to do with product rule, but I can't figure out how exactly it was used

6. (Original post by funkiichiicka)
Thank you.

I thought it was something to do with product rule, but I can't figure out how exactly it was used

You're welcome.

We know the product rule for differentiation is the following.

Replace v with X and the required result follows.

I hope that helps.

Darren

P.S.: The centre dot denotes product. It isn't necessary here, but some people choose to use it.
7. (Original post by funkiichiicka)

Thank you.

I thought it was something to do with product rule, but I can't figure out how exactly it was used

is the function multiplied by the function . We can just product rule it:

8. (Original post by DPLSK)
You're welcome.

We know the product rule for differentiation is the following.

Replace v with X and the required result follows.

I hope that helps.

Darren

P.S.: The centre dot denotes product. It isn't necessary here, but some people choose to use it.
Aah!!! Got it! Thank you so much!

Thank you for the help. Much appreciated.

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