You were talking about proofs of irrationality. However pi and e are known to be transcendental, and the proof of transcendence is usually harder than a proof of irrationality. Differential equations is indeed an indispensable tool in proving transcendence. Look at any book which has transcendental numbers in its title.

The key authors are Baker, Mahler, Shidlovski, Gelfond and Siegel.

However, if we are talking about the proofs of IRRATIONALITY (that is of numbers which were not known to be irrational), here is a famous ans relatively simple proof:

Apéry, Roger Irrationalité de $\zeta(2)$ et $\zeta(3)$. Astérisque 61, 11-13 (1979).

Then you can look at the papers which cite this great result.

You were talking about proofs of irrationality. However pi and e are known to be transcendental, and the proof of transcendence is usually harder than a proof of irrationality. Differential equations is indeed a useful tool in proving transcendence. Look at any book which has transcendental numbers in its title.

The key authors are Baker, Mahler, Shidlovski, Gelfond and Siegel.

However, if we are talking about the proofs of IRRATIONALITY (that is of numbers which were not known to be irrational), here is a famous and relatively simple proof:

Apéry, Roger Irrationalité de $\zeta(2)$ et $\zeta(3)$. Astérisque 61, 11-13 (1979).

Then you can look at the papers which cite this great result.