Consider the initial value problem for a Riccati equation:
Find the true solution y(x). How many solutions have you found?
This is actually part of a project involving us researching the riccati equation. So, I haven't learned anything about this topic in class, so I'm really not sure how to begin.
I've found guide on the website http://www.sosmath.com/diffeq/first/...i/riccati.html but I'm not sure how reliable it is.
Only for clarification purpose: the 'initial value problem' to be 'attacked' is this...
... or some different?...
Sorry, I copy and pasted and the subscripts and superscripts came out wrong. It should be:
All right!... the Riccati equation is written as...
In general the solution of (1) is possible only if you know one particular solution and that is in most cases an hard problem. The equation you have proposed however is...
... and, because the independent variable x doesn't partecipate, we can use a different approach. Let's suppose in (2) to swap the x and y, so that the equation becomes...
In this case you can separate the variables, integrate immediately and obtain...
The constant c can be obained with the condition and that gives . The (4) is easily invertible, so that the 'true solution' of (2) is...