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September 21st, 2017, 05:35 PM  #1 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0  Nonlinear Nonseparable first order differential equation
Find the general solution of y' = (t + y  1)^2 and write it in explicit form. This is clearly a nonlinear, nonseparable 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. 
September 21st, 2017, 06:42 PM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,610 Thanks: 550 Math Focus: Yet to find out. 
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. 
September 21st, 2017, 08:19 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,299 Thanks: 1688 
Let u = t + y  1, then u' = 1 + y' = 1 + u², which is separable.

September 22nd, 2017, 06:41 AM  #4 
Senior Member Joined: Oct 2015 From: Antarctica Posts: 128 Thanks: 0 
Yeah, that's the way to do it alright. Just not very intuitive, but that definitely works.

September 22nd, 2017, 08:04 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,344 Thanks: 2466 Math Focus: Mainly analysis and algebra 
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.

September 22nd, 2017, 10:07 PM  #6  
Senior Member Joined: Sep 2015 From: USA Posts: 2,042 Thanks: 1064  Quote:
$y^\prime = (y+t1)^2$ 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 $u = y+t1$ 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.  
September 23rd, 2017, 04:27 AM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,344 Thanks: 2466 Math Focus: Mainly analysis and algebra  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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