Welcome to the first module! We begin by introducing differential equations and classifying them. We then explain the Euler method for numerically solving a first-order ode. Next, we explain the analytical solution methods for separable and linear first-order odes. An explanation of the theory is followed by illustrative solutions of some simple odes. Finally, we present three real-world examples of first-order odes and their solution: compound interest, terminal velocity of a falling mass, and the resistor-capacitor electrical circuit.

We begin by generalising the Euler numerical method to a second-order equation. 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 convert the 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 discuss the solutions for these different cases. We then consider the inhomogeneous ode, and the phenomena of resonance, where the forcing frequency is equal to the natural frequency of the oscillator. Finally, some interesting and important applications are discussed.

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 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.