Verified Certificate

Supported by the following organizations

This is to certify that Michael Ludwig 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

Sanjay Sarma

Sanjay Sarma

Vice President for Open Learning Massachusetts Institute of Technology

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About edX Verified Certificates

An edX Verified Certificate signifies that the learner has agreed to abide by the edX honor code and completed all of the required tasks of this course under its guidelines, as well as having their photo ID checked to verify their identity.