In the Beginning...

Hello!

When I first began studying ordinary differential equations (ODEs), I immediately fell in love with their partial counterparts. To me, it felt as though the study of ODEs was a simple set up to understanding partial differential equations (PDEs) sometime in the near future. As I got further into the subject of ODEs, however, I began to experience how difficult all the theoretical groundwork to solving them really was. It all got to a point where we were studying only certain, ideal cases of these equations because it so happened that the vast majority of them had either no analytical solutions or required some arcane numerical methods to solve. All throughout this journey, every once in a while, I would remember PDEs and cringe at the thought of having to solve one of those, but I really wanted to, nevertheless, especially after learning that they are a cutting-edge subject still in its theoretical infancy.

In my study of ODEs, I learned how to solve equations of the form$$\sum_{k=0}^{n}a_k(x)\frac{d^ky}{dx^k}=b(x)$$by means of countless methods akin to recipes in a cookbook. Perhaps this was due to the nature of the course seeking to instill the idea in the form of mathematical euphemisms, to put it that way. For example, to solve the ODE$$\frac{dy}{dx}=e^{3x+2y}$$we must:

(1) recognize that it is separable and separate$$\frac{dy}{dx}=(e^{3x})(e^{2y})\Longrightarrow e^{-2y}dy=e^{3x}dx$$(2) integrate and 'absorb' integrating constants$$\int e^{-2y}dy=\int e^{3x}dx\Longrightarrow-\frac{1}{2}e^{-2y}=\frac{1}{3}e^{3x}+C$$(3) solve for $y$$$y=-\frac{1}{2}\ln\left(-\frac{2}{3}e^{3x}+C\right)$$But how can we be sure that this method works? That is, how do we know that this is the solution? Is it unique? Or, in other words, could there be others?

An even more outrageous 'recipe' to solving an ODE is the idea of an integrating factor, which claims that an ODE of the form$$\frac{dy}{dx}+P(x)y=f(x)$$can be solved by multiplying everything by the integrating factor $e^{\int P(x)dx}$. For example, solve$$3\frac{dy}{dx}+12y=4.$$(1) get into standard form$$\frac{dy}{dx}+4y=\frac{4}{3}$$(2) find the integrating factor$$e^{\int4dx}=e^{4x}$$(3) multiply everything by it$$e^{4x}\frac{dy}{dx}+4e^{4x}y=\frac{4}{3}e^{4x}$$(4) wrap it all up by factoring, integrating and solving for $y$$$\begin{align}

\frac{d}{dx}\left[e^{4x}y\right]&=\frac{4}{3}e^{4x}\\

e^{4x}y&=\frac{1}{3}e^{4x}+C\\

y&=Ce^{-4x}+\frac{1}{3}

\end{align}$$What in the world? Who came up with that method, where is the theorem that supports it, and can it be applied to other differential equations?

Another thing that blew my mind was the Principle of Superposition, which claims that if an ODE has two solutions $y_1$ and $y_2$, then their sum $y_1+y_2$ is another solution. Why is that true!? It makes total sense when you look at it from a physical point of view (think the Doppler effect), but where is the mathematical groundwork?

There are many other methods or 'recipes' to solve ODEs, like substitutions, exact and Bernoulli forms, numerical methods, Laplace transforms and Fourier series, to name a few, but most of them were presented to us in the form of a toolkit obtained by rote memorization.

Now that I am studying PDEs, we began our journey by studying the convoluted Heat Equation. There are many things that I have accumulated about it thus far, but that will be the subject of another entry. All I will say right now is that my most dreaded nightmare has become true; PDEs are magnitudes harder!

In a good way, however, and profound enough that we just 'get' the main idea behind the inspiration of all the mathematical behemoths who pioneered their theoretical development. It is no longer a toolkit approach, but one based solely on maieutics channeled toward our deepest, analytic conscience. Perhaps back then I was not ready to undergo such rigorous treatment of this kind of equations, but now I feel just a tad more ready to give it one heck of a good try.

Stay tuned!