Solve Riccati Equation: dy/dx + y2 = 1 + x2

[STEMucator]

Homework Statement

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

Homework Equations

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.