Ordinary Differential Equations (Math 2280)

At Utah State University, Math 2280 covers basic theory and methodology behind solving ordinary differential equations.  This page is a summary of most topics covered in Ordinary Differential Equations (Math 2280) with the chapters and content closely aligned with the textbook "Ordinary Differential Equations" by Kenneth Howell.  This is a list that summarizes derivative, integration and identity rules.

Chapter 6 • Substitutions

Just like substitutions were used in previous experiences with calculus, various expressions (usually referred to as u) can be defined which simplify the process of solving the differential equation. Common substitutions include \(u = \frac{y}{x}\) and \(u = y^{1-n}\) (where n is the power in a Bernoulli equation).

Chapter 7 • Exact Equations

A special form of a differential equation is one where it is of the form \(M(x,y) = N(x,y)\frac{dy}{dx} = 0\). It can be inferred that this equation (or some multiple of it using an integrating factor \(\mu\)) is such that there is some equation \(\phi\) so that \(\phi_y = M\) and \(\phi_x = N\). That is, the elements of the differential equations are simply the partial derivatives of some other function. Then, one of them can be integrated independent of the other (such as integrating \(M\) with respect to  \(x\)) and then use this to solve for \(\phi\).  

Chapter 8 • Slope Fields

One way to gain some understanding of the behavior of a differential equation is by looking at a slope field. This plots a small line segment at each \((x,y)\) point showing the instantaneous slope at that point. This can indicate visually some characteristics, such as the constant solutions and whether or not they are stable.

Chapter 10 • Modeling

Some of the most useful applications of differential equations come from using them to model real-life scenarios. For example, looking at how an object heats/cools over time, or how the concentration of substances in various liquids vary as time progresses. If modeled correctly, the same skills that have been used to solve other differential equations can be applied to those situations. However, sometimes the solutions may not make sense in the context of the original situation.

Chapter 11 • Beginning Second-order Equations

This chapter is intended as an introduction to equations involving \(\frac{d^2}{dx^2}\), also known as second order. Much time in the book is spent looking at various ways to solve these, but it begins by using the method of using a substitution \(v = \frac{dy}{dx}\). This can essentially reduce the order of the equation to a first order equation in terms of \(v\) so it can be solved (for \(v\)) and then integrated so \(y\) can be determined.

Chapter 12 • Reduction of Order

The ideas of chapter 11 are expanded to recognize substitution involving lower order derivatives can be applied to various situations, where again it can be solved in terms of the new variable using the same methods as before, then integrated to find the original. Note that this can be applied to \(n^{th}\) order equations, sometimes with repeated substitution. When one of the solutions to the equation is already known (usually given as part of the problem or easily deduced), it can be assumed that the set of all solutions will be some \(u\) times the other solution. Therefore, one can take the derivative(s) of this and substitute it into the differential equation so it can be solved in terms of \(u\) and then integrated to find the solution in terms of \(y\). Another way to think about this is to recognize that \(u = \frac{y}{y_1}\), where \(y_1\) is the known solution.