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## Homework Statement

Consider the differential equation [itex]y'=x-y^2[/itex]. Find maxima, minima and critical points; show that for every solution [itex]f=f(t)[/itex] exists [itex]T\geq 0[/itex] such that [itex]f(t)< \sqrt{T}\;\forall t > T[/itex]

## Homework Equations

The Riccati equation: [itex]y'=a(x)y^2+b(x)y+c(x)[/itex]

The Bernoulli equation: [itex]y'=a(x)^n+b(x)y[/itex]

## The Attempt at a Solution

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I've been trying to study the differential equation given by [itex]y'=x-y^2[/itex] 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 [itex]y'=a(x)y^2+b(x)y+c(x)[/itex] using [itex]c(x)=x, a(x)=-1, b(x)=0[/itex]

and is possible to get a general solution if I know two other solutions: if [itex]f_1,f_2[/itex] are solutions then [itex]f=f_1+C(f_2-f_1)[/itex] 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 it in the general case.