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.