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Are there general methods to solve a differential equation in this form:

$$\frac{\mathrm{d}y}{\mathrm{d}t}= F(x(t), y(t))\frac{\mathrm{d}x}{\mathrm{d}t} \\ y(t_0) = y_0,\quad \\ x(t_0) = x_0 \quad $$

Is there a name for such equations, considering that the function $F$ is not really amenable to separation of variables?

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    $\begingroup$ Well, you can remove $t$ from the problem to get $dy/dx=F(x,y)$, but after that it is a completely generic first order equation. $\endgroup$ – Ian Jan 8 '18 at 22:39
  • $\begingroup$ Oh yes. That makes sense. I don't know why i didn't think of that. $\endgroup$ – Hikaru Jan 8 '18 at 22:43

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