Verified Certificate

Supported by the following organizations

This is to certify that K RAGAVAN successfully completed and received a passing grade in 18.031x: Introduction to Differential Equations a course of study offered by MITx, an online learning initiative of the Massachusetts Institute of Technology.

Noted by

David Jerison

David Jerison

Professor Department of Mathematics Massachusetts Institute of Technology

Krishna Rajagopal

Krishna Rajagopal

Dean for Digital Learning Massachusetts Institute of Technology

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