Quasi-homogeneous 1st Order ODEs
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Quasi-homogeneous 1st Order ODEsWondering 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
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Quasi-homogeneous 1st Order ODEsThe LHS is the product rule for differentiation. It is necessary to find(Original post by funkiichiicka)
I understand where the RHS came from... but how did they get the LHS of that?
.
It was used to give us(Original post by funkiichiicka)
What happened to the 1 and -4 we found earlier?
.
I hope that helps.
DarrenLast edited by DPLSK; 15-04-2012 at 20:56. -
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Re: Quasi-homogeneous 1st Order ODEs(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
Please can you explain
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Re: Quasi-homogeneous 1st Order ODEsYou're welcome.(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
Please can you explain
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.Last edited by DPLSK; 15-04-2012 at 21:32. -
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Re: Quasi-homogeneous 1st Order ODEs(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
Please can you explain
is the function 
multiplied by the function 
. We can just product rule it:
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Re: Quasi-homogeneous 1st Order ODEsAah!!! Got it! Thank you so much!(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.
Thank you for the help. Much appreciated.
