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Nonlinear first order DE 
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#1
Jul1008, 10:01 AM

P: 11

Hello:
I discovered this forum while looking for advice on solving a first order nonlinear differential equation. The equation I am trying to solve is dy/dx=(3ay+3bx^2y^2)/(3xbx^3y) a and b are constants. The equation is not exact, nor is it homogeneous. I have failed to separate the variables by factoring. So the usual methods don't work. Any help or advice will be appreciated. 


#2
Jul1008, 04:37 PM

P: 135

Are you sure it isn't homogeneous?



#3
Jul1008, 05:05 PM

P: 11

The equation is not homogeneous. See if you can find a work around. Thanks 


#4
Jul1008, 05:52 PM

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P: 26,148

Nonlinear first order DE
(are you sure it isn't (3axbx^3y) on the bottom? anyway …) Hint: first, factor it out as much as you can, then make the obvious substitution. 


#5
Jul1108, 09:04 AM

P: 11

It is 3x and not 3ax. I am going to try an iterative approach. Nothing else seems to work. Thanks 


#7
Jul1108, 05:45 PM

P: 11

I have tried factoring the equation  but no luck! Can you help with the factoring? Thanks 


#8
Jul1108, 06:14 PM

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P: 26,148

= (3y/x)(a + bx^{2}y)/(3  bx^{2}y) What's difficult about that? Now make the obvious substitution … 


#9
Jul1208, 08:29 AM

P: 11

So substitution will not help. 


#10
Jul1208, 10:22 AM

P: 443

I think he meant to substitute new variable x^2 y(x) = f(x) and then separation of variables x and f works.



#11
Jul1208, 12:24 PM

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P: 26,148

Exactly! (btw, have you noticed the new x^{2} and x_{2} tags on the Reply to thread page? ) But obviously it doesn't actually solve the problem. In hindsight, what part of "Now make the obvious substitution" did you not think worth trying? Anyway, as smallphi suggests, put z = x^{2}y … what is dz/dx? 


#12
Jul1208, 12:45 PM

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P: 12,016

To give you a few further hints:
xy=z/x, and y/x=z/(x^3). 


#13
Jul1208, 05:03 PM

P: 11

I tried the substitution; I do get a result even if looks horrible!
I will repeat the calculation, just to make sure. Thanks guys! 


#14
Jul1208, 07:03 PM

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P: 26,148

It shouldn't look horrible. What dz/dx did you get? 


#15
Jul1308, 08:16 AM

P: 11

This is what I get dz/dx=(z/x)((3a+b)+z(32b)/(3bz)) Solving for z gives a bunch of ln terms. 


#16
Jul1308, 09:01 AM

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P: 26,148




#17
Jul1308, 10:09 AM

P: 11

But, it is a solution! If you have a simpler expression I would like to see it. Much appreciate your interest and effort. 


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