This course is all about differential equations and covers both theory and applications. In the first five weeks, students will learn about ordinary differential equations, while the sixth week is an introduction to partial differential equations.

# Differential Equations for Engineers

This course is part of Mathematics for Engineers Specialization

Taught in English

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Instructor: Jeffrey R. Chasnov

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## Course

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## What you'll learn

First-order differential equations

Second-order differential equations

The Laplace transform and series solution methods

Systems of differential equations and partial differential equations

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31 quizzes

## Course

(2,073 reviews)

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## There are 6 modules in this course

A differential equation is an equation for a function with one or more of its derivatives. We introduce different types of differential equations and how to classify them. We then discuss the Euler method for numerically solving a first-order ordinary differential equation (ODE). We learn analytical methods for solving separable and linear first-order ODEs, with an explanation of the theory followed by illustrative solutions of some simple ODEs. Finally, we explore three real-world examples of first-order ODEs: compound interest, the terminal velocity of a falling mass, and the resistor-capacitor electrical circuit.

#### What's included

14 videos14 readings7 quizzes

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. Using 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.

#### What's included

11 videos11 readings4 quizzes1 plugin

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.

#### What's included

12 videos9 readings5 quizzes

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 into it here, an introduction to this technique may be useful to students who encounter it again in more advanced courses.

#### What's included

11 videos10 readings5 quizzes

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 then visualized using phase portraits. We next 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. We then apply the theory to solve a system of two coupled harmonic oscillators, and use the normal modes to analyze the motion of the system.

#### What's included

13 videos10 readings5 quizzes

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 describing the diffusion of a dye in a pipe. We then proceed to solve this PDE using the method of separation of variables. This involves dividing the PDE into two ordinary differential equations (ODEs), which can then be solved using the standard techniques of solving ODEs. We then use the solutions of these two ODEs, and our definition of a Fourier series, to recover the solution of the original PDE.

#### What's included

11 videos11 readings5 quizzes

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The Hong Kong University of Science and Technology

The Hong Kong University of Science and Technology

Korea Advanced Institute of Science and Technology(KAIST)

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