My Math Forum  

Go Back   My Math Forum > College Math Forum > Differential Equations

Differential Equations Ordinary and Partial Differential Equations Math Forum


Thanks Tree9Thanks
  • 3 Post By Joppy
  • 3 Post By skipjack
  • 2 Post By v8archie
  • 1 Post By romsek
Reply
 
LinkBack Thread Tools Display Modes
September 21st, 2017, 06:35 PM   #1
Senior Member
 
Joined: Oct 2015
From: Antarctica

Posts: 103
Thanks: 0

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 09:21 PM.
John Travolski is offline  
 
September 21st, 2017, 07:42 PM   #2
Senior Member
 
Joined: Feb 2016
From: Australia

Posts: 1,411
Thanks: 481

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 09:06 PM.
Joppy is offline  
September 21st, 2017, 09:19 PM   #3
Global Moderator
 
Joined: Dec 2006

Posts: 18,155
Thanks: 1422

Let u = t + y - 1, then u' = 1 + y' = 1 + u², which is separable.
skipjack is offline  
September 22nd, 2017, 07:41 AM   #4
Senior Member
 
Joined: Oct 2015
From: Antarctica

Posts: 103
Thanks: 0

Yeah, that's the way to do it alright. Just not very intuitive, but that definitely works.
John Travolski is offline  
September 22nd, 2017, 09:04 AM   #5
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,034
Thanks: 2342

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.
Thanks from greg1313 and topsquark
v8archie is online now  
September 22nd, 2017, 11:07 PM   #6
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: Southern California, USA

Posts: 1,604
Thanks: 817

Quote:
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.
Thanks from v8archie
romsek is offline  
September 23rd, 2017, 05:27 AM   #7
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,034
Thanks: 2342

Math Focus: Mainly analysis and algebra
Quote:
Originally Posted by romsek View Post
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.
v8archie is online now  
Reply

  My Math Forum > College Math Forum > Differential Equations

Tags
differential, equation, nonlinear, nonseparable, order



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Can't solve this First Order Linear Differential Equation Addez123 Calculus 0 May 5th, 2017 07:13 PM
Differential Equation with Separable Variables Help Valheru Differential Equations 2 May 28th, 2015 08:20 AM
First order linear partial differential equation alithebig Differential Equations 0 May 26th, 2013 07:19 PM
Second order non-linear differential equation JulieK Differential Equations 2 December 22nd, 2012 08:41 AM
Second Order Linear Differential Equations Sefrez Differential Equations 1 October 10th, 2011 08:10 AM





Copyright © 2017 My Math Forum. All rights reserved.