Contact Info
Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Phone: 519-888-4567, ext. 32700
Fax: 519-746-4319
PDF files require Adobe Acrobat Reader
Differential equations are important because for many physical systems, one can, subject to suitable idealizations, formulate a differential equation that describes how the system changes in time. Understanding the solutions of the differential equation is then of paramount interest.
The simplest ODE is:
You can verify that any function
y(t) = Cekt
where C is a constant, is a solution to this ODE. If k > 0, this ODE thus describes exponential growth (for example, unrestricted population growth). If k < 0, the interpretation is exponential decay (for example, the decay of radioactive atoms).
In many applications you will encounter the ODE:
where ω is a constant.
You can verify that any function
y(t) = A sin(ωt + ø) ,
where A and ø are constants, is a solution. This ODE thus describes a physical system whose behaviour is periodic in time. We say that the system undergoes simple harmonic motion with amplitude A, frequency ω, and phase ø.
In the foundational courses on ODEs you will learn to identify whether an ODE can be solved explicitly (most cannot!), or whether it is necessary to solve it numerically using a computer.
Contact Info
Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Phone: 519-888-4567, ext. 32700
Fax: 519-746-4319
PDF files require Adobe Acrobat Reader
The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.