# Thread:

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!
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9100672602653503, "perplexity_flag": "head"}