Calculus 2 Lecture 8.1: Solving First Order Differential Equations By Separation of Variables

The speaker introduces the topic of differential equations, clarifying that the session will not be a comprehensive class on the subject but a basic introduction. They discuss the relevance of differential equations in engineering and various applications, emphasizing the importance of understanding the concept of 'order' in differential equations. The speaker also explains the difference between differential equations with unknowns in the derivative and those with functions of the derivative, providing an example of a first-order differential equation and how it can be solved by integration.

The speaker delves into first-order differential equations that involve functions of both variables, X and Y, and highlights the challenges they present. They introduce the concept of separable differential equations as a simpler subset of these equations, where the variables can be separated for easier solving. The speaker reassures the audience that despite the complex terminology, the process of solving separable differential equations is manageable and builds upon previously learned concepts.

The speaker explains what constitutes a separable differential equation, focusing on the ability to separate the variables into a product or a sum. They provide examples of both separable and non-separable differential equations, illustrating the distinction between those that can be solved using separation of variables and those that cannot. The speaker also discusses the process of solving separable differential equations by integrating both sides of the equation with respect to the same variable.

The speaker provides a step-by-step guide on solving separable differential equations, starting with ensuring the equation fits the criteria of being separable. They demonstrate the process of moving all terms involving one variable to one side and integrating both sides with respect to that variable. The speaker uses an example involving the natural logarithm and exponential functions to illustrate the integration process and the simplification of the resulting equation.

The speaker discusses the use of initial conditions in differential equations, explaining how they allow for the determination of the arbitrary constant in the general solution. They differentiate between a general solution, which is a family of curves, and a specific solution, which is a single curve that passes through a given point. The speaker demonstrates how to apply an initial condition to find the specific solution to a differential equation.

The speaker continues the discussion on solving differential equations with initial values, emphasizing the process of finding the specific solution by substituting the initial values into the general solution to solve for the arbitrary constant. They reiterate the importance of having a general solution to plug in the numbers and show how the initial value problem helps in narrowing down to a single curve from the family of curves represented by the general solution.

The speaker introduces the concept of natural growth and decay models, which describe the change in a population or quantity over time based on its current size. They explain that these models are based on the principle of proportional change, where the rate of change is directly proportional to the current state. The speaker sets the stage for solving differential equations related to growth and decay by separation of variables.

The speaker develops a model for natural growth and decay by solving a first-order differential equation that represents the rate of change of a population. They demonstrate the process of separating variables and integrating both sides with respect to time. The resulting model is an exponential function that includes an arbitrary constant, which can be determined using an initial condition.

The speaker explains how to use an initial condition to solve for the arbitrary constant in the growth and decay model. They emphasize that the arbitrary constant represents the starting population or initial amount of the substance, and by plugging in the initial condition, one can determine its specific value. The speaker also highlights the importance of understanding the context of the model, such as the difference between growth and decay scenarios.

The speaker discusses the law of cooling, which describes how an object cools over time in relation to the ambient temperature. They provide a personal anecdote about Thanksgiving and how the law of cooling can be applied to understand the rate at which food cools. The speaker emphasizes the importance of understanding the difference between the object's temperature and the ambient temperature in the cooling process.

The speaker provides a detailed explanation of how to calculate the temperature of an object over time using the law of cooling. They discuss the differential equation that models this process and demonstrate the steps to solve it by separation of variables. The speaker then shows how to use initial conditions to find a particular solution to the differential equation, which can be used to predict the temperature of the object at any given time.

The speaker concludes the discussion by addressing the practical question of how long one must wait to cut a turkey without burning their fingers. They use the particular solution derived from the law of cooling to calculate the time it takes for the turkey's temperature to drop to a safe level. The speaker demonstrates the process of plugging in the desired temperature into the model and solving for the time, providing a practical application of the concepts discussed.