What Is Required for Preparation for Differential Equations?

[Bogrune]

At the moment, my first choice for a major is in Economics, and after doing some research, I've found out that the mathematics courses required for a PhD are Probability and Statistics, Single-Variable Calculus (Calc. 1 and 2), Multivariable Calculus, Linear Algebra and both Ordinary and Partial Differential Equations. My question is: What is required for preparation for Differential Equations? Also, are Partial Differential Equations really as difficult as most people say they are (which they is are very difficult)?

 

[clope023]

At the moment, my first choice for a major is in Economics, and after doing some research, I've found out that the mathematics courses required for a PhD are Probability and Statistics, Single-Variable Calculus (Calc. 1 and 2), Multivariable Calculus, Linear Algebra and both Ordinary and Partial Differential Equations. My question is: What is required for preparation for Differential Equations? Also, are Partial Differential Equations really as difficult as most people say they are (which they is are very difficult)?

You need Calc II and Linear Algebra for ordinary differential equations, pde's will require those and multivariable calculus; it can be very difficult depending on the professor and whether he cares or not that you can finish your exams on time.

 

[czelaya]

I took a two semester part course in ordinary differential equations course, and like the previous post stated the prereqs depend on the instructor or orientation of the course. The full sequence of calculus and multivariable calculus were the prerequisites for the course that I took. We used some linear algebra but what was needed was taught in class (very basic concepts like the transpose, determinant, inverse matrix, and so forth).

Differential equations reminded so much of organic chemistry(a course offered in chemistry departments that tends to weed out chemistry and molecular biology majors), in the sense, that it requires you to exhaustively exercise a lot of what you learned in previous courses. Once you understand what differential equations are, the solutions to the DE's are usually a recipe of steps that you utilize to find the solutions. Along the way you learn new mathematics like basic integral transforms such as Laplace transforms and gamma functions to aid in finding solutions.

When I took partial differential equations, the concepts were basically(primarily because PDE's deal with more than one variable) the same as ordinary differential equations but the recipes, in most cases(well at least for the course that I took), were different. The solutions either required some type of methods of approximation(Power series, Taylor polynomials, ect) and the very powerful tools of Fourier analysis.

 

[snipez90]

For a top PhD program in econ, having a grasp on real analysis will be pretty helpful. The more math you can handle, the better. For instance, at UChicago, graduate admissions filter your undergrad courses into math and econ categories.

More to the point though, I think understanding mathematical analysis will be more helpful than pursuing ODE and then PDE. Analysis gives you the tools to study ODE/PDE deeply, and to understand the ramifications and limitations behind the methods used in DE courses. Proper understanding of probability theory and game theory also requires a good grasp on real analysis. But even if you take a look at a basic microecon grad text such as Varian, you'll see the ideas of analysis pop up.

Now to strictly answer your question, PDE doesn't seem that important to have for PhD admissions, unless you're planning on getting a PhD in Math Finance. It's certainly unlikely to show up in basic graduate sequences in econometrics, price theory, micro. Differential equations do show up in economic growth, but a good understanding of ODEs should suffice for the foundations of the subject.

Finally, a basic computationally-focused course in PDE will not be that difficult provided you have a good grasp on ODEs. PDE is a very deep subject, but there are very common methods for dealing with linear PDEs. These include the method of characteristics, Fourier transforms, and separation of variables, none of which is particularly difficult to grasp. Only the first of these few methods really requires a basic range of multivariable calculus techniques. However, to fully grasp basic linear PDE theory (e.g. issues of convergence, regularity, etc.), you must be familiar with basic graduate analysis (i.e. measure theory) and vector calculus.

In any case, taking multivariable calculus is a good idea, because basic undergrad micro/macro involves solving consumer choice optimization problems. The basic tool here is Lagrange multipliers, which properly belongs under MV calc. However, you can pick up on basic MV calc on your own and can spend your credits on other crucial courses such as real analysis.

 

[Bogrune]

Thanks for the advice everyone! I was thinking of taking both Single and Multi-Varible Calculus because I was told by my Precalculus instructor that those classes will be crucial for nearly all the other mathematics courses I'll be taking.

More to the point though, I think understanding mathematical analysis will be more helpful than pursuing ODE and then PDE. Analysis gives you the tools to study ODE/PDE deeply, and to understand the ramifications and limitations behind the methods used in DE courses. Proper understanding of probability theory and game theory also requires a good grasp on real analysis. But even if you take a look at a basic microecon grad text such as Varian, you'll see the ideas of analysis pop up.

Mathematical Analysis eh? Does that also include complex analysis? I'm just asking because though Mathematical Analysis looks a bit difficult, I'm willing to take those courses if I really need them, and most importantly if it's fun to work with!

Though I'm undecided what I want to major in, Economics is my first choice at the moment because I enjoyed it when I took A.P. Economics in my senior year in High School, and because it's pretty close to Business and Finance. Mainly, because I read that it requires strong knowledge in mathematics, and not something as simple as Intermediate Algebra and Statistics.

 

[snipez90]

Analysis is difficult, but once you grasp real analysis, you will have strengthened your intuition for calculus concepts. There are times, especially in the beginning, when analysis may not seem like a whole lot of fun. But as with any subject, you have to learn gritty basics before you can get anywhere.

Basic complex analysis is fairly easy to grasp once you've done real analysis. The methods employed in complex analysis are actually very different from those in real analysis. It's a beautiful subject, but I'm not aware of any applications to economics. However, one rather early application of complex analysis is computing very difficult definite integrals.

You'll have to do a lot of algebra in undergrad econ courses. In any consumer optimization problem, for example, setting up the problem and using lagrange multipliers is maybe 20% of the work. The rest is slogging through tedious algebra to obtain the optimal points.

Once you get to econometrics, economic growth, game theory, and more advanced coursework, things get more interesting. But it turns out you probably still have to do a lot of algebra, though you won't have to solve the same optimization problem over and over.

Fixed point theorems show up often in economics (e.g. to establish equilibrium points). This is yet another application of analysis, or rather topology.