Differential equations provide a framework for describing processes in terms of rates of change of their components. For example, a bathtub filled with a concentrated solution of dye X can be drained at the same rate as pure water is introduced. Over time, the concentration of dye X goes down. This we can determine intuitively, but we can also model the process with a differential equation, an equation with derivatives as variables, i.e. with dy/dt dependent on y. Here, "solving" the differential equation gives us y in terms of t in simple terms. In the example I have given, it turns out y decays exponentially with time. The neat thing about differential equations is that we can model a system entirely in terms of rates (which are intuitive to grasp). Using calculus, we can then deduce mechanically from those rates the exact behavior of the system.
Unfortunately, solving a differential equation is not always easy. This is where MATH 215 comes into the picture. As a first course in differential equations, MATH 215 introduces the key concepts and methods necessary to solve first and second order linear ordinary differential equations (ODEs). As my instructor put it, this actually constitutes 0% of all possible differential equations, but it gives us excellent tools for modeling systems nonetheless.
Systems often depend upon more than one variable. For example, an extension of the simple "Newton's Cooling" ODE presented in first year mathematics courses is the Heat Equation. Here, heat dissipates along a direction, x, and its "diffusion" is observed through time, t. This requires a description that have rates depending independently on two different variables, requiring partial derivatives. Such a differential equation is called a partial differential equation (PDE). This is the subject of MATH 316.
An Outline of MATH 215
I divide MATH 215 into five distinct sections
- Introduction, first order ODEs, and their applications (Midterm 1).
- Second order ODEs and their applications (Midterm 2).
- The Laplace transform (Midterm 3).
- Systems of First Order Linear ODEs (Midterm 3).
- Qualitative description of nonlinear systems of ODEs.
The first two sections deal with solving ODEs directly using calculus-based methods. First order ODEs are solved using separation of variables, integrating factors, and other such techniques. Second order ODEs come in many varieties, and are treated in a series of different cases and using specially derived theorems.
The third section covers a different method altogether for solving ODEs, in which the ODE is transformed into different variables that allow it to be solved using purely algebraic methods (including LOTS of partial fractions) and then transformed back.
In the fourth section, linear algebra is used in order to solve systems of first order ODEs. This was tricky at first, because it had been over a year since MATH 223, but I soon found it very clear. Linear algebra knowledge is a must for this, but rarely does the course go beyond 2x2 matrices. The last section worked on the ability to draw vector fields formed by ODEs freehand without solving the ODE in detail. It turns out that many ODEs that cannot be solved in this course can nonetheless be drawn. I found this section to be the most fascinating of the course.
Assignments were given out weekly and typically took me around 10 hours to complete on average. There were 3 midterms. Between the 3 midterm grades and 9 of 11 assignment grades, one of the four is discarded in the calculation of a final grade. The three remaining average out to give 45% of the final grade. Students must pass the assignment component in order to pass the course. The remaining 55% came from the final exam. Please note that these details are relevant to the course as I took it, and may change depending on the instructor. They serve only to give a general idea of structure and grading.
An outline of Math 316
Math 316 starts with a review of ODEs. It then introduces one more powerful method for solving ODEs: the series solution. Using this method, ODEs and their solutions are represented in terms of power series. This is a much more general approach for solving ODEs and can solve many problems that previously introduced methods cannot.
After this, there was a brief section on solving partial differential equations using computational/numerical methods. To keep things simple, these methods were carried out on Microsoft Excel. This involves representation of PDE solutions in discrete rather than continuous form. The excel templates for this section were provided online, and we only needed to modify them to solve the desired problems.
The remaining two thirds of the course dealt in analytic solutions to PDEs. This was first performed mechanically using algebraic and calculus-based methods, and then generalized into eigenvalue/eigenfunction problems under the heading of "Sturm-Liouville" theory, which I found to be the most challenging part of the course.
My experience with these courses
The theory behind differential equations is dense and detailed. However, for physicists and engineers, they serve their purposes mainly by their applications. As a result, these courses can be taught in two ways: (1) by exploring the detailed mathematical theory, or (2) by focusing on the applications and problem-solving aspects of the course. For me, MATH 215 was taught in the former way, and MATH 316 was taught in the latter. Naturally, MATH 215 turned out to be very difficult, and MATH 316 very easy for me. Both were taught excellently and clearly, however, which may account for my fascination with differential equations and consequently the ridiculous length of this post.
MATH 215 was very time-consuming, in particular its assignments. To be honest, the course terrified me. I did not understand much in the first section because it was densely theoretical and I hadn't taken a math course since my first term of first year. I choked on my first midterm exam and received 50%. Luckily, my grade was salvageable, but this would mean I had to complete assignments diligently and ace both remaining midterm exams. This was the first term I seriously considered a minor in physics, and it all hinged on my performance in MATH 215.
I found that the course got clearer and more interesting as it progressed. It became a balancing act between understanding theory and carrying out procedural calculations. As my interest piqued, so did my motivation, and this ended up being my second highest mark of first and second year. With this encouragement, I happily declared a minor in Physics.
Studying for MATH 215 required a thorough review of all concepts of the course, as well as assignments, textbook problems, and past final exams. I also found the course on ODEs from MIT Open Courseware extremely helpful. It is a great course to look at for a preview on MATH 215 or as a supplement. Link: http://www.youtube.com/watch?v=XDhJ8lVGbl8. The professor, Dr. George Bluman, was a key asset. His treatment of the content promoted thorough and conceptual understanding. In addition, his devotion to student learning was clear, and his end of term review sessions were incredibly helpful. If he is your instructor, I strongly recommend seeing him during office hours or review sessions as he is genial, patient, and clear when helping to explain difficult problems and concepts.
The following term, I took MATH 316, and found it thoroughly easy in comparison to MATH 215. The course was less theory driven and focused more on calculations. Assignments took about half the time as in MATH 215. Again, this was likely a product of teaching style rather than course content.
Closing statement
In the term I took MATH 215, I also took CHEM 205, CHEM 211, and BIOL 201. An understanding of differential equations gave me a deeper understanding of chemical kinetics in these courses, not to mention a facility with mathematical reasoning. Differential equations constitute a critically important field in mathematics. Their theory brings concepts of calculus into clear focus and their applications to fields throughout science and engineering are countless. It is difficult to get by in the study of physics without a thorough understanding.
MATH 215 was very time-consuming, in particular its assignments. To be honest, the course terrified me. I did not understand much in the first section because it was densely theoretical and I hadn't taken a math course since my first term of first year. I choked on my first midterm exam and received 50%. Luckily, my grade was salvageable, but this would mean I had to complete assignments diligently and ace both remaining midterm exams. This was the first term I seriously considered a minor in physics, and it all hinged on my performance in MATH 215.
I found that the course got clearer and more interesting as it progressed. It became a balancing act between understanding theory and carrying out procedural calculations. As my interest piqued, so did my motivation, and this ended up being my second highest mark of first and second year. With this encouragement, I happily declared a minor in Physics.
Studying for MATH 215 required a thorough review of all concepts of the course, as well as assignments, textbook problems, and past final exams. I also found the course on ODEs from MIT Open Courseware extremely helpful. It is a great course to look at for a preview on MATH 215 or as a supplement. Link: http://www.youtube.com/watch?v=XDhJ8lVGbl8. The professor, Dr. George Bluman, was a key asset. His treatment of the content promoted thorough and conceptual understanding. In addition, his devotion to student learning was clear, and his end of term review sessions were incredibly helpful. If he is your instructor, I strongly recommend seeing him during office hours or review sessions as he is genial, patient, and clear when helping to explain difficult problems and concepts.
The following term, I took MATH 316, and found it thoroughly easy in comparison to MATH 215. The course was less theory driven and focused more on calculations. Assignments took about half the time as in MATH 215. Again, this was likely a product of teaching style rather than course content.
Closing statement
In the term I took MATH 215, I also took CHEM 205, CHEM 211, and BIOL 201. An understanding of differential equations gave me a deeper understanding of chemical kinetics in these courses, not to mention a facility with mathematical reasoning. Differential equations constitute a critically important field in mathematics. Their theory brings concepts of calculus into clear focus and their applications to fields throughout science and engineering are countless. It is difficult to get by in the study of physics without a thorough understanding.
