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In ordinary calculus, one can solve a function by taking its "anti-derivative," in a form of integration.

Likewise, in differential equations, one can look for solutions by integrating the equation in some way.

Is this analogous to "taking an anti-derivative?" Does the process have a name, perhaps "anti-differential?" And is this made possible (as in the case of taking an anti-derivative) by the fundamental theorem of calculus?

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First to answer your question, "No." Solving a DE is simply called "solving a DE".

To see why this might be consider the following analogy between solving DEs and solving polynomial equations:

Think: integration corresponds to root taking. So for example, the simplest DEs are ones of the form $y^{(n)}=f(x)$ where the solution just involves integrating $n$-times. Likewise the simplest polynomial equations are ones of the form $x^n=a$ where the solution just involves computing the $n$-root of $a$.

If you have a more complicated polynomial equation, say $x^2+3x+1=0$, square roots [, cube roots, etc.] alone won't allow you to find the roots of this equation. Instead a series of additions, subtractions, multiplications, divisions, and root extractions are required to find the roots.

The same goes for more complicated DEs. Integration will only get you so far. And there are many examples of DEs whose solutions are unobtainable using "algebraic" methods and integration alone.

If you're interested in sort of thing, there is an area of mathematics called "Galois Theory" which allows one to understand exactly when algebraic methods allow one to find roots of polynomials. [The non-existence of a "quintic formula" (analogous to the quadratic, cubic, and quartic formulas) was what gave birth to Galois theory in the first place.] In the same vein, "Differential Galois Theory" is the theory of when one can "solve" (in a certain technical sense) DEs. Be warned both of these fields require a lot of background to understand.

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A differential equation is not necessarily a single derivative, but rather more typically a sum of derivatives of different orders, possibly also multiplied together. So the solution of a differential equation is not really an "anti-differential" operation, but rather more generally is an "inverse method": a method that seeks to compute the original function that gives rise to the differential equation in the first place.

Some differential equations can be integrated in the traditional sense, and in such a case you are correct in a sense. The FTC certainly comes into play.

But not every differential equation can be solved by an integration operation, or a series of integration operations. In fact, existence and uniqueness of solutions is not generally guaranteed.

Nevertheless, if you think of integration as a process that takes an equation that contains some non-zero derivative order of a function, and yielding a relation that reduces that derivative order, then solving a differential equation (when a solution exists) is like doing enough integration steps to completely remove all derivative orders. It's almost never that neat, however.

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Maybe I am a bit nihilistic, but it's just terminology. Since the most basic ODE is $$y'(x)=f(x),$$ which is solved by $$y(x)=\int f(x)\, dx,$$ we currently say "integrate an ODE" to mean that we want to solve it. This expression does not imply that a solution will be immediately found by an integration in the ordinary sense. We also speak of the "general integral" of an ODE to mean the most general solution to that ODE. I think you should not take the verb "to integrate" literally, in the framework of differential equations. Just imagine that some integral will come into play, sooner or later.

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That's fair enough.+1 –  Tom Au Sep 7 '12 at 14:17

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