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 describesÌýexponential growthÌý(for example, unrestricted population growth). If k < 0, the interpretation isÌýexponential decayÌý(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 isÌýperiodic in time. We say that the system undergoesÌýsimple harmonic motionÌýwith 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.