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I'm trying to think of a function $x(t)$ such that $\dot{x}(t) = x(t)^2t.$ I worked out for my self that it has something to do with the $x(t) = -2t^{-2}$ but I couldn't work out how to put the constant in so that the function still worked.

Since then I've been told that the function is $x(t) = \frac{2}{c-t^2}$ and I can easily see that this function works but I can't work out how to get to that answer without just having to spot it. Is there a way to work this out or does one just need to be good at spotting these things?

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    http://en.wikipedia.org/wiki/Separation_of_variables2012-05-14

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We solve this by separation of variables:

You have $\frac{dx}{dt} = x(t)^2t$and hence that $\frac{dx}{x(t)^2} = t dt$. Integrating both sides you get that $-x(t)^{-1} = \frac{t^2}{2} + C$ for some constant $C$, from which it follows that $x(t) = \frac{-1}{\frac{t^2}{2} + C}$

and hence that $x(t) = \frac{2}{c - t^2}$ for $c = -2C$. We can write it like this since $c,C$ are just arbitrary constans. This is the general solution of your ODE above.

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There are many methods for solving differential equations. It depends on the type of the equation you have. For this particular example, the equation can be written as:

$ \frac{\dot{x}(t)}{x(t)^2} = t $

Which is of the form:

$ \frac{\dot{x}}{h(x)} = g(t) $

Equations that can be written like this are called separable. To solve, simply integrate both sides:

$ \int \frac{\dot{x}(t)}{x(t)^2} \, dt = \int t \, dt $

Or:

$ \int \frac{dx}{x^2} = \int t \, dt $

Can you take it from there? It should be a straightforward integration problem.