From the course: Complete Guide to Differential Equations Foundations for Data Science

Initial value problems

From the course: Complete Guide to Differential Equations Foundations for Data Science

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Initial value problems

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- [Instructor] Let's wrap up this chapter with exploring initial value problems. You may be wondering why I am discussing these elusive initial value problems when you should be focusing on differential equations. This is because they are highly integrated with differential equations. If you remember from the previous video, a particular solution represents a specific solution to a differential equation. With particular solutions, remember there are no arbitrary constants in the solution because the constants are solved utilizing initial conditions by plugging them into the differential equation. The format of the problem presented to find a particular solution is actually an initial value problem. An initial value problem must consist of a differential equation and at least one initial condition. The goal of solving an initial value problem is to find a particular solution that satisfies the initial condition or conditions that have been provided. You've heard me mention the word initial conditions a few times, so let's take a moment to understand exactly what they are. Initial conditions are values of the solution to a differential equation and/or its derivatives at specified points. For example, an initial condition may be Y of zero equals five, or it might be Y prime of three equals two divided by five. A quick note is that initial value problems are not the same as boundary value problems despite their similar names. Initial value problems focus on one point for each derivative level versus boundary value problems look at two or more points. Boundary value problems are more complex than initial value problems, and they will be thoroughly explored in a chapter at the end of this course. You can have initial value problems for any order of a differential equation. A general rule is the number of initial conditions must equal the order of the differential equation. So for example, if you have a second order equation, then there must be two initial conditions corresponding with that. The general process for solving an initial value problem is step one, you'll find the general solution to the differential equation. Step two, you'll apply initial conditions with the general solution to determine constant values. And finally, in step three, you will verify the solution satisfies the differential equation and any initial conditions. Another term you should know is interval of validity. The interval of validity is the largest possible range of values where the solution to a differential equation and its initial condition or conditions is valid. This range must be defined and continuous. Note that this range can also contain all real numbers from negative infinity to positive infinity. Let's look at an example of an initial value problem. Let's say you have the differential equation Y prime equals three multiplied by E to the X plus X squared minus four, and then you have an initial condition of Y of zero equals seven. This is an initial value problem because you have your differential equation and at least one initial condition. If you solve this and use your initial condition, you will then get a particular solution of Y equals three multiplied by E to the x plus 1/3 multiplied by X cubed minus four, multiplied by X+4. You will learn in later chapters how to actually solve this type of equation step by step, but this should give you a general idea of how to identify what an initial value problem is in the first place, and again, the goal to find that particular solution. Initial value problems are key in mathematics because they're used in theorems such as the existence and uniqueness theorem. This is also known as the Picard-Lindelof Theorem. In biology, they are used to model population growth. In engineering, they are used with motion under forces, and finally, in chemistry, they are used with chemical reactions. This should provide you a solid understanding of initial value problems. Be prepared to see these problems heavily utilized throughout the rest of this course. In future chapters, you'll dive into detail on what methods you can use to solve these types of problems for a variety of differential equations. This wraps up your introduction to differential equations, but note that this is just the beginning. I suggest you take time to thoroughly understand these basic concepts since they will be heavily used throughout the rest of this course. In the next chapter, you will learn all about first order differential equations.

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