# Thread: Riccati Equation: existence/uniqueness relation

1. ## Riccati Equation: existence/uniqueness relation

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

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

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

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

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

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

Since both $f(x,y)$ and $\frac{\partial f}{\partial y}$ are continous on a rectnagle containing (0,1) the solution exists and is unique.

3. 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

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

Thanks!