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Say you had the differential equation

$$\frac{\mathrm{d}x}{\mathrm{d}t} = f(x,t)$$

Separating variables, you get $$\mathrm{d}x = f(x,t)\,\mathrm{d}t$$

From there, when I try to solve for $x$, I integrate both sides

$$x = \int f(x,t)\,\mathrm{d}t$$

Without an explicit function, how do you continue to solve this?

I've seen differentiation under the integral sign, which says

$$\frac{\mathrm{d}}{\mathrm{d}t}\int f(x,t)\,\mathrm{d}t = \int\frac{\partial}{\partial t}f(x,t)\,\mathrm{d}t$$

but would that mean I'd need to differentiate both sides, and I'd end up with $\frac{\mathrm{d}x}{\mathrm{d}t}$ on the left side again?

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  • $\begingroup$ Why do you think there's a better way of writing this? $\endgroup$ – user223391 Sep 17 '15 at 4:51
  • $\begingroup$ I guess technically I don't assume there is. I am just curious. $\endgroup$ – galois Sep 17 '15 at 4:52
  • $\begingroup$ Even with some explicit functions, it would be impossible to provide a closed form. Well-known example: $f(x,t)=e^{t^2}$. $\endgroup$ – Quang Hoang Sep 17 '15 at 4:54
  • $\begingroup$ In general there is no way to continue past this point. You are stuck unless you know the function. $\endgroup$ – Brevan Ellefsen Sep 17 '15 at 4:56
  • $\begingroup$ @BrevanEllefsen, so at this point it just stays left at $x=\int f(x,t)\,\mathrm{d}x$ ? $\endgroup$ – galois Sep 17 '15 at 4:57
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There's no nice way to write this, what you have is the best way to write this. To see why it's so hopeless, I offer a much easier problem. Given a function $f(x,t)$, can you write a general form for

$$\frac{\partial}{\partial t}f(x,t)$$

If you can't even find a formula for this, integration, which is much harder, is even more hopeless.

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