Riccati equations

1. Oct 7, 2012

Zondrina

1. The problem statement, all variables and given/known data

So I came across one of these equations. Solve : dy/dx + y2 = 1 + x2 when y(x) = x is a particular solution.

2. Relevant equations

3. The attempt at a solution

So I re-arranged my equation into Riccati form : dy/dx = 1 + x2 - y2

Now I let : v = $\frac{1}{y-y(x)}$ so that y = x + 1/v

Thus : dy/dx = -(1/v2)dv/dx + 1

Subbing these back into my equation yields :

(-1/v2)dv/dx + 1 = 1 + x2 - (1/v + x)2

Simplifying everything, I get a seperable linear equation in standard form :

dv/dx - 2xv = 1 (***)

So my integrating factor μ must satisfy :

dμ/dx = -2xμ which implies that μ = e-x2

So now my equation (***) can be re-written as :

d/dx [e-x2v] = e-x2

Integrating both sides and then expressing my integrand in terms of the error function gives me the final answer :

v = $\frac{\sqrt{\pi} \space erf(x) + d}{2e^{-x^2}}$

And finally plugging v back into y yields my solution :

y = $x + \frac{2e^{-x^2}}{\sqrt{\pi} \space erf(x) + d}$

Is this correct? This is my first try at one of these so I'm still in the not sure what I'm doing moment.

Last edited: Oct 7, 2012