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?

  • $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.