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I've been trying to study the differential equation given by $y'=x-y^2$ for a while, and I didn't was even close to a solution. Finally when I gave up I solved this equation with Wolfram Alpha, and it seems that the solution was quite intricate and doesn't seem possible to solve only with elementary functions, see Wolfram's solution.

Now my question is, what are the possible methods to use when I encounter a non-linear equation of like this?, is it possible to say something about it without the need of a computer?.

One thing I saw is that this is similar the Riccati equation $y'=a(x)y^2+b(x)y+c(x)$ using $c(x)=x, a(x)=-1, b(x)=0$ and is possible to get a general solution if I know two other solutions: if $f_1,f_2$ are solutions then $f=f_1+C(f_2-f_1)$ is a general solution. The problem here is that I'd need to find particular solutions for the equations, something that I couldn't figure out how to do in the general case.

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The equation has two nullclines $x=\pm\sqrt{t}$, which is where the slope field is horizontal. Important features of these nullclines:

  1. $x=\sqrt{t}$ begins at $(0,0)$ and increases to $\infty$
  2. $x=\sqrt{t}$ has positive slope everywhere; $x=-\sqrt{t}$ has negative slope everywhere.
  3. Solutions have negative slope above $x=\sqrt{t}$ and below $x=-\sqrt{t}$. They have positive slope between them.

Using the above properties, the following observations can be made.

  • Putting 1 and 3 together, one can conclude that every solution with positive initial value will decrease until it crosses the upper nullcline at some moment $t=t_0$.

  • From 2 one can infer that solutions do not cross the upper nullcline in the upward direction, and do not cross the lower nullcline in the downward direction. Therefore, once they get into the region between the nullclines, they stay there.

  • From 3, solutions increase as long as they are between the nullclines.

  • Also from 3: if a solution starts below the lower nullcline (negative initial value) it will just continue to decrease.

Conclusions

  • A solution with positive initial value has one minimum (where it crosses the upper nullcline) and no maxima.

  • A solution with zero initial value is always increasing.

  • A solution with negative initial value is either always decreasing (if it never crosses the lower nullcline), or has one minimum (if it crosses the lower nullcline).

Also see my answer to Properties of the solutions to $x'=t-x^2$.

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