In 1953 Ilyin obtained his Candidate of Science degree in Physics and Mathematics for the thesis «Diffraction of electromagnetic waves on some inhomogeneities», his scientific advisor being Andrey Tikhonov.

In 1958 he obtained Doctor of Science degree in Physics and Mathematics for his thesis «On convergence of expansions in eigenfunctions of Laplace operator».

Ilyin was the author of more than 140[1] research papers and 17 monographs on mathematical analysis, analytical geometry, and linear algebra, which were published both in Russia and abroad.

He was also the member of Scientific and Methodological Council on Mathematics under the Ministry of Education of Russia.

His son, Aleksandr Ilyin, the Corresponding Member of the Russian Academy of Sciences, is a professor of the Department of Nonlinear Dynamic Systems and Control Processes at CMC MSU.

The lecture courses he gave within his pedagogical activity included: «Equations of Mathematical Physics», «Equations of Elliptic Type», «Functional Analysis», «Mathematical Analysis», and «Linear Algebra and Analytical Geometry».

In the late 1960s Ilyin developed a universal method that made it possible for an arbitrary selfadjoint second-order operator in an arbitrary (not necessarily bounded) domain to establish the final conditions of uniform (on any compact) convergence for both spectral expansions themselves and their Riesz means in each of the classes of functions (Nikolsky, Sobolev-Liouville, Besov and Sigmund-Holder function classes).

In 1972 he published a negative solution to the problem posed by Sergei Sobolev on the convergence for

He developed a new method for estimating the remainder term of the spectral function of an elliptic operator in both the metric

Ilyin made a fundamental contribution to the spectral theory of nonself-adjoint operators.

He obtained the conditions under which the system of eigenvectors and associated vectors for the one-dimensional boundary value problem has the basis property in

Malkov in 1989, he demonstrated that the previously established conditions for the basis property of the system of eigenfunctions and associated functions of an operator

For a number of cases, he obtained formulas describing optimal boundary controls (in terms of minimizing the boundary energy) that transfer the system from a given initial state to a given finite state (the results obtained in co-authorship with Evgeny Moiseev are among the best achievements of the Russian Academy of Sciences in 2007 year).