Re: help with solving Riccati equation

word.drunk@xxxxxxxxx writes:

here's the problem:
show that y1(x) = 1/x is a solution of the riccati equation y'=1/(x^2)
- y/x - y^2

and then find the general solution of y'=1/(x^2) - y/x - y^2

im learning this from the book and whatever other sources i can find.

i started solving it by

since y = 1/x => y = 1/x + 1/z => dy/dx = 1/(x^2) - 1/(z^2) dz/dx


z? What z?

Showing that y1=something is a solution of a differential equation is simply a
matter of plugging in y1 for y in the differential equation and doing whatever
simplification is necessary to get left and right sides to agree.

A trick that sometimes works for getting the general solution from a
particular solution y1(x) is to try y(x) = u(x) y1(x). You may find out that the
differential equation for u(x) that you get by substituting this into your
Ricatti equation is something you can solve.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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