This course is about differential equations and covers material that all engineers should know. Both basic theory and applications are taught. In the first five weeks we will learn about ordinary differential equations, and in the final week, partial differential equations. The course is composed of 56 short lecture videos, with a few simple problems to solve following each lecture. And after each substantial topic, there is a short practice quiz. Solutions to the problems and practice quizzes can be found in instructor-provided lecture notes. There are a total of six weeks in the course, and at the end of each week there is an assessed quiz.
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What You Will Learn
- First-order differential equations
- Second-order differential equations
- The Laplace transform and series solution methods
- Systems of differential equations and partial differential equations
Syllabus
WEEK 1
First-Order Differential Equations
A differential equation is an equation for a function with one or more of its derivatives. We introduce differential equations and classify them. We then learn about the Euler method for numerically solving a first-order ordinary differential equation (ode). Then we learn analytical methods for solving separable and linear first-order odes. An explanation of the theory is followed by illustrative solutions of some simple odes. Finally, we learn about three real-world examples of first-order odes: compound interest, terminal velocity of a falling mass, and the resistor-capacitor electrical circuit.
WEEK 2
Homogeneous Linear Differential Equations
We generalize the Euler numerical method to a second-order ode. We then develop two theoretical concepts used for linear equations: the principle of superposition, and the Wronskian. Armed with these concepts, we can find analytical solutions to a homogeneous second-order ode with constant coefficients. We make use of an exponential ansatz, and transform the constant-coefficient ode to a second-order polynomial equation called the characteristic equation of the ode. The characteristic equation may have real or complex roots and we learn solution methods for the different cases.
WEEK 3
Inhomogeneous Linear Differential Equations
We now add an inhomogeneous term to the constant-coefficient ode. The inhomogeneous term may be an exponential, a sine or cosine, or a polynomial. We also study the phenomena of resonance, when the forcing frequency is equal to the natural frequency of the oscillator. Finally, we learn about three important applications: the RLC electrical circuit, a mass on a spring, and the pendulum.
WEEK 4
The Laplace Transform and Series Solution Methods
We present two new analytical solution methods for solving linear odes. The first is the Laplace transform method, which is used to solve the constant-coefficient ode with a discontinuous or impulsive inhomogeneous term. The Laplace transform is a good vehicle in general for introducing sophisticated integral transform techniques within an easily understandable context. We also introduce the solution of a linear ode by a series solution. Although we do not go deeply here, an introduction to this technique may be useful to students that encounter it again in more advanced courses.
WEEK 5
Systems of Differential Equations
We learn how to solve a coupled system of homogeneous first-order differential equations with constant coefficients. This system of odes can be written in matrix form, and we learn how to convert these equations into a standard matrix algebra eigenvalue problem. The two-dimensional solutions are visualized using phase portraits. We then learn about the important application of coupled harmonic oscillators and the calculation of normal modes. The normal modes are those motions for which the individual masses that make up the system oscillate with the same frequency.
WEEK 6
Partial Differential Equations
To learn how to solve a partial differential equation (pde), we first define a Fourier series. We then derive the one-dimensional diffusion equation, which is a pde for the diffusion of a dye in a pipe. We proceed to solve this pde using the method of separation of variables.
