Differential Equations for Engineers

A differential equation is an equation for a function with one or more of its derivatives [1][2]. 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, terminal velocity of a falling mass, and the resistor-capacitor electrical circuit.

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.

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.

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.