Differential Equations: Lecture 1.1-1.2 Definitions and Terminology and Initial Value Problems

The script begins with an introduction to differential equations (DEs), explaining that they are equations involving an unknown function and its derivatives. The presenter illustrates this with the example of a simple DE, y'' + y = 0, and its solution in the form of a linear combination of cosine and sine functions. The context of DEs in various scientific fields, including physics, chemistry, biology, and finance, is briefly touched upon, with a mention of the famous Black-Scholes equation. The importance of DEs is emphasized, leading into a discussion about the types of DEs: ordinary differential equations (ODEs) and partial differential equations (PDEs), with an example given for each type.

The second paragraph delves into the concept of the order of a DE, which is defined as the highest derivative present in the equation. The presenter uses several examples to clarify this, such as first-order DEs with first derivatives and higher-order DEs with second or even fourth derivatives. The explanation is accompanied by interactive questioning with the audience, aiming to solidify their understanding of how to identify the order of a DE.

This section introduces the distinction between linear and nonlinear DEs. The presenter provides a formal definition of a linear DE in the form of an equation where the function y and its derivatives appear to the first power with pure functions of x as coefficients. The explanation is supplemented with examples to demonstrate how to distinguish between linear and nonlinear DEs, emphasizing the complexity and difficulty in solving nonlinear DEs compared to their linear counterparts.

The script discusses various types of solutions to DEs, such as one-parameter and two-parameter families of solutions, which are sets of solutions containing arbitrary constants resulting from integration. The concept of a particular solution is introduced as a specific member of the family of solutions, determined by specific values of the arbitrary constants. The trivial solution, where y equals zero, is highlighted as a special case. The importance of understanding these different types of solutions is underscored for solving DEs effectively.

The explanation continues with the difference between explicit and implicit solutions. Explicit solutions are those where y is expressed directly in terms of x, while implicit solutions are defined by an equation where y is not isolated. The presenter provides examples to illustrate both types, emphasizing the importance of recognizing whether a solution is explicit or implicit for further analysis and application.

The concept of the interval of definition is introduced as the largest interval over which a solution to a DE is defined. The presenter demonstrates how to determine this interval using a function and its graph. Additionally, singular solutions are discussed as solutions that cannot be obtained by selecting values for arbitrary constants within a family of solutions. The presenter uses examples to highlight how singular solutions can be overlooked and their importance in solving DEs.

This section focuses on initial value problems (IVPs), which are DEs accompanied by a condition that specifies the value of the solution at a particular point. The presenter explains how IVPs are used to select a particular solution from a family of solutions, using the initial condition to determine the value of any arbitrary constants involved. The graphical interpretation of IVPs is also discussed, showing how the condition helps in identifying the specific solution curve that passes through the given point.

The script provides a step-by-step walkthrough of solving an IVP, including finding the particular solution and determining the interval of definition. The presenter uses a nonlinear DE and an initial condition to demonstrate the process of plugging in values to find the specific solution that fits the condition. The importance of understanding the interval of definition for the initial condition is also emphasized, with examples showing how to choose the correct interval based on the given condition.

The final paragraph addresses the challenging scenario where an IVP does not yield a solution when the initial condition cannot be satisfied by the given family of solutions. The presenter explains that despite not finding a particular solution that fits the condition, it is still necessary to determine the interval of definition. The concept of singular solutions is revisited, and the presenter demonstrates how to identify and work with such solutions, especially when they are the only solutions that satisfy the DE and the initial condition.