Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

General and particular solution for this first-order nonlinear ODE :

$$y'(x)+\frac{1}{x}=\frac{1}{y}$$

share|cite|improve this question
    
And the stated problem is to prove something about the solution, or actually to write it down explicitly? – GEdgar Dec 15 '12 at 17:44

$y'(x)+\dfrac{1}{x}=\dfrac{1}{y}$

$y\dfrac{dy}{dx}+\dfrac{y}{x}=1$

This belongs to an Abel equation of the second kind.

Let $x=e^{-t}$ ,

Then $\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}=\dfrac{\dfrac{dy}{dt}}{-e^{-t}}=-e^t\dfrac{dy}{dt}$

$\therefore-e^ty\dfrac{dy}{dt}+e^ty=1$

$y\dfrac{dy}{dt}-y=-e^{-t}$

This belongs to an Abel equation of the second kind in the canonical form.

Please follow the method in https://arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.