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# How do I solve this differential equation? watch

1. dw/dt = (a + bw) w^2

I separated the variables and then used partial fractions but everything got too complicated and I end up with two 'log (w)' terms and a 'w' term on the LHS and t on the RHS. I can't rearrange that to w=[something in terms of t]. So may be I'm doing it wrong.

(e.g. I got log (f(w)) + k(w) = t + c) where f and k are different linear functions of w.)

Thanks
2. (Original post by VA11ZZY)
dw/dt = (a + bw) w^2

I separated the variables and then used partial fractions but everything got too complicated and I end up with two 'log (w)' terms and a 'w' term on the LHS and t on the RHS. I can't rearrange that to w=[something in terms of t]. So may be I'm doing it wrong.

(e.g. I got log (f(w)) + k(w) = t + c) where f and k are different linear functions of w.)

Thanks
You won't be able to find the expression explicitly. Can you show us how you did the partial fraction?

I end up with (checked with Wolfram for various values of a and b):
Spoiler:
Show

3. (Original post by Blazy)
You won't be able to find the expression explicitly. Can you show us how you did the partial fraction?

I end up with (checked with Wolfram for various values of a and b):
Spoiler:
Show

Yes - Wolfram Alpha gives that answer - the problem, as you pointed out, is rearranging the expression to give w in terms of t.

Someone on stack exchange gave this answer (leaving out the constants for simplicity)

solves to

And then using the substitution of

it rearranges to

because Lambert_W(−exp(c−1−t)) = v

The Lambert function appears to be a numerical function which gives the solution to Y = Xe^X. Now I just need to figure out how to use it in this context!
4. (Original post by VA11ZZY)

Yes - Wolfram Alpha gives that answer - the problem, as you pointed out, is rearranging the expression to give w in terms of t.

Someone on stack exchange gave this answer (leaving out the constants for simplicity)

solves to

And then using the substitution of

it rearranges to

because Lambert_W(−exp(c−1−t)) = v

The Lambert function appears to be a numerical function which gives the solution to Y = Xe^X. Now I just need to figure out how to use it in this context!
I see. Assuming what you've written is true, we can pull off a similar trick then. It'll be hard to do this without hints but basically:

1) and stick into our constant of integration.

2) Make the substitution

Then work your way through for a similar answer, remembering the basic log rules - I think this will work.

EDIT: Just realised where that substitution comes from. Making the substitution directly works fine (without doing step 1).

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