
Riccati Equation
Consider the initial value problem for a Riccati equation:
$\displaystyle y' + y^{2} + 1 = 0, y(0) = 1$
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...
$\displaystyle y^{'}= y^{2} + y + 1$ , $\displaystyle y(0)=1$
... or some different?...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$

Sorry, I copy and pasted and the subscripts and superscripts came out wrong. It should be:
$\displaystyle y' + y^{2} + 1 = 0, y(0) = 1$

All right!... the Riccati equation is written as...
$\displaystyle y^{'}= p(x) + q(x)\cdot y + r(x)\cdot y^{2}$ (1)
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...
$\displaystyle y^{'}= (1+y^{2})$ , $\displaystyle y(0)=1$ (2)
... 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...
$\displaystyle x^{'}= \frac{1}{1+y^{2}}$ , $\displaystyle x(1)=0$ (3)
In this case you can separate the variables, integrate immediately and obtain...
$\displaystyle x=  \tan^{1} y + c$ (4)
The constant c can be obained with the condition $\displaystyle x(1)=0$ and that gives $\displaystyle c=  \frac {\pi}{4}$. The (4) is easily invertible, so that the 'true solution' of (2) is...
$\displaystyle y= \tan (x  \frac{\pi}{4})$ (5)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$