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September 21st, 2017, 05:35 PM   #1
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Exclamation Non-linear Non-separable first order differential equation

Find the general solution of
y' = (t + y - 1)^2
and write it in explicit form.

This is clearly a non-linear, non-separable first order differential equation, but I'm really struggling; I don't know what method to use to solve this! What do you recommend?

Last edited by skipjack; September 21st, 2017 at 08:21 PM.
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September 21st, 2017, 06:42 PM   #2
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This is Riccati's equation, a special case of the Bernoulli equation. Have you studied that?

Last edited by skipjack; September 21st, 2017 at 08:06 PM.
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September 21st, 2017, 08:19 PM   #3
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Let u = t + y - 1, then u' = 1 + y' = 1 + u², which is separable.
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September 22nd, 2017, 06:41 AM   #4
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Yeah, that's the way to do it alright. Just not very intuitive, but that definitely works.
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September 22nd, 2017, 08:04 AM   #5
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Spotting substitutions is almost as important in solving differential equations as it is in solving integrals. If that one isn't intuitive to you, I suggest you practice more.
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September 22nd, 2017, 10:07 PM   #6
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Originally Posted by John Travolski View Post
Yeah, that's the way to do it alright. Just not very intuitive, but that definitely works.
This is how I'd look at this problem.

$y^\prime = (y+t-1)^2$

as you noted it's not separable.

well what is separable? If you think like I do you'd come up with

$u^\prime = u^2$

and you can easily separate this into

$\dfrac{du}{u^2} = dt$

well given the form of the original equation what will $u$ have to be?

it's pretty clear that $u = y+t-1$

well ok let's try this and see what we get

$\dfrac{du}{dt} = \dfrac{dy}{dt} + 1$

$\dfrac{dy}{dt} = \dfrac{du}{dt} -1$

$u^\prime -1=u^2$

$u^\prime = u^2 + 1$

etc.

so sometimes you just have to try things out and see if they work.
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September 23rd, 2017, 04:27 AM   #7
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sometimes you just have to try things out and see if they work.
This. In spades. Mathematics isn't about being to solve everything first time just by looking at it. It's about trying out the mathematical tools you have learned (or sometimes not). It's about applying techniques creatively to see what helps.
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