Modelling with first-order differential equations

Define differential equation: A differential equation is an equation involving derivatives. The order of a differential equation is determined by the highest derivative present. Therefore, a first-order differential equation contains only first derivatives.
Understand how to solve first-order differential equations: The standard form of a first-order differential equation is: dy/dx = f(x,y). To solve such equations, one usually separates the variables, divides the equations into two parts pertaining to y and x, then integrate.
Recognition of separable variables: It is important to recognize when an equation is separable, meaning it can be written in the form dy/dx = g(x)h(y). This form allows you to separate the variables and integrate.
Understand the relationship between differential equations and physical phenomenon: Most physical phenomena can be expressed in terms of differential equations. Astronomy, optics, heat conduction, and electricity and magnetism are areas where differential equation models are often used.
Knowing homogeneous differential equations: If the differential equation dy/dx = f(x,y) can be transformed to the type dy/dx = g(y/x) or dy/dx = g(x/y), the equation is said to be homogeneous.
Be familiar with integrating factors: An integrating factor is a function that is chosen to facilitate the solution of a given equation, particularly a linear first-order differential equation with variable coefficients.
Understand the application and interpretation of initial value problems: An initial value problem involves finding a particular solution to a differential equation which meets a specific condition at a specified value of the independent variable.
Recognize and solve exact differential equations: An inexact differential equation can sometimes be turned into an exact differential equation using an integrating factor.
Know how first-order differential equations are used in a wide range of contexts to model situations where one quantity is changing with respect to another.
Ability to interpret the solution to differential equations in context, including recognising equilibrium solutions and interpreting their stability.
Understand the Euler’s method: Euler’s method provides a numerical approximation to the solutions of a first-order differential equation when analytical solutions are difficult to find.
Often, evaluating the behavior of solutions as they approach infinity, or the behavior at specific points, is a crucial part of the overall problem-solving strategy.