Differential equations are, in addition to a topic of study in mathematics, the main language in which the laws and phenomena of science are expressed. In basic terms, a differential equation is an expression that describes how a system changes from one moment of time to another, or from one point in space to another.
When working with differential equations, the ultimate goal is to move from a microscopic view of relevant physics to a macroscopic view of the behavior of a system as a whole.
Let’s look at a simple differential equation. Based on previous math and physics courses, you know that a car that is constantly accelerating in the x-direction obeys the equation d2x/dt2 = a, where a is the applied acceleration. This equation has two derivations with respect to time, so it is a second-order differential equation; because it has derivations with respect to only one variable (in this example, time), it is known as an ordinary differential equation, or an ODE.
Let’s say that we want to solve the above ODE for the position of the car as a function of time. We can do so by using direct integration: the integration of both sides with respect to time gives us dx/dt = at + c, where c is a constant of integration. If the velocity of the car is known to be a particular value at some point in time T, we can solve for c as c = [dx/dt]t=T / aT. More simply, if the velocity is zero at time 0, then c = 0. Integrating again gives us the desired solution: x(t) = at2/2 + ct + e, where e is another constant of integration. Again, if the position of the car at t=0 is taken to be zero, then the solution for the position of the car becomes x(t) = at2/2. It is useful to note that checking the validity of a solution to an ODE is easily accomplished by substituting it back into the ODE.
Unfortunately, not all differential equations are this easy to solve. Generally, an ODE is a functional relation (it would be a function, except that the “variables” are themselves functions!) between an independent variable t, a dependent function U(t), and some of its derivatives diU(t)/dti. An ODE is linear if it can be written as a functional relation in which no powers of U or its derivatives appear—otherwise, the ODE is nonlinear. For the most part, nonlinear ODEs can only be solved numerically; this course will focus on linear ODEs.
This course will also introduce several other subclasses and their respective properties. However, despite centuries of study, the only practical approach to the solution of complicated ODEs that has emerged is numerical approximation. Although these numerical techniques are the subject of numerical analysis courses (see MA213: Numerical Analysis), this course will introduce you to the fundamentals behind numerical solutions.
Upon successful completion of this course, students will be able to:
- Identify ordinary differential equations and their respective orders.
- Explain and demonstrate how differential equations are used to model certain situations.
- Solve first order differential equations as well as initial value problems.
- Solve linear differential equations with constant coefficients.
- Use power series to find solutions of linear differential equations.
- Solve linear systems of differential equations with constant coefficients.
- Use the Laplace transform to solve initial value problems.
- Use select methods of numerical approximation to find solutions to differential equations.
