Riccati Equation: existence/uniqueness relation

• Apr 11th 2009, 10:50 AM
tn2k7
Riccati Equation: existence/uniqueness relation
What does the existence/uniqueness theorem say about solutions to initial value problems for Riccati equations of the form

http://www.mathhelpforum.com/math-he...24257514-1.gif

This is what I came up with, but I'm not sure.

A Riccati Equation of a particular form has a unique solution satisfies the following conditions:
It is to exist for large values of the independent variable

Its graph must stay above a certain line for large values of the variable
• Apr 11th 2009, 12:06 PM
TheEmptySet
Quote:

Originally Posted by tn2k7
What does the existence/uniqueness theorem say about solutions to initial value problems for Riccati equations of the form

http://www.mathhelpforum.com/math-he...24257514-1.gif

This is what I came up with, but I'm not sure.

A Riccati Equation of a particular form has a unique solution satisfies the following conditions:
It is to exist for large values of the independent variable

Its graph must stay above a certain line for large values of the variable

First we need to isolate the derivative

$\displaystyle \frac{dy}{dx}=f(x,y)$ so we get

$\displaystyle f(x,y)=y'=-(y^2+1)$

Since both $\displaystyle f(x,y)$ and $\displaystyle \frac{\partial f}{\partial y}$ are continous on a rectnagle containing (0,1) the solution exists and is unique.
• Apr 12th 2009, 01:58 PM
tn2k7
Hi, thanks for your help. I was wondering if you (or anyone else) could help me with this similar problem.

What does the existence/uniqueness theorem say about solutions to initial value problems for Riccati equations of the form

$\displaystyle y'(x) + y(x)^{2} + Ay(x) + B = 0.$

Thanks!