Solution to differential equation $y'(x) = a * y(x)^2$

Question

first of all: I am not a mathematician. I am struggling since a few hours with a simple differential equation which I would like to solve to approximate the expectation curve for computer simulations I am doing.

Recursive function:

$$y[x+1] = y[x] - c * y[x]^2$$

If c is small, this can be approximated by the following differential equation:

$$y'[x] = -c * y[x]^2$$

Which I do not manage to solve...

Sorry for such a low-level question here and thanks :)

EDIT: I know that $y$ at position $x$=0 equals 0.8.

Answer 1

We have

$$\frac{dy}{dx} = -cy^2$$ Rearranging: $$\frac{dy}{y^2}=-cdx$$ Integrating both sides: $$-\frac{1}{y}=-cx+d$$ for some constant of integration $d$. Rearranging: $$y=\frac{1}{cx-d}$$

Answer 2

You should solve the integral:

$$\int y^{-2}dy = -\int cdx$$

and you will get a solution to this aproximation.