Differential equations came paired with the invention of calculus and have been around for a long time. They are still the main study of a lot of mathematicians. Many famous mathematicians such as Newton, Leibniz, d’Alembert, Euler and Lagrange have made discoveries in differential equations. Differential equations find their use in areas such as Physics, Biology, Chemistry and Economics. In this article I will dive into the world of differential equations, go over the main discoveries that have been made, and provide a better understanding of them.
Types of differential equations
The first type of differential equation that most students of mathematics encounter is the first order ordinary differential equation. Examples of types of ordinary differential equations are separable ODEs, linear ODEs with constant coefficients and systems of ODEs. These types of first order ODEs have been studied extensively and are all relatively easy to solve. When moving to higher order ODEs we are able to solve some second order nonlinear differential equations using a method called reduction of orders. Easier higher order ODEs are the linear ODEs with constant coefficients. In all these types we can consider both homogeneous and inhomogeneous ODEs. More on the difference between homogeneous and inhomogeneous later.
Moving on to PDEs, the classifications are a bit different. Here we first look at whether the PDE is homogeneous or inhomogeneous. A PDE has a dependent variable often denoted by $u$ and a vector of independent variables often denoted by $x_1, x_2,…,x_n$ or $x, t$. A PDE is then called homogeneous if all terms in the differential equation depend on the dependent variable $u$ or its derivatives. If there is one or more terms that do not depend on $u$ or its derivatives the PDE is called inhomogeneous. Examples of these will be given later. After you have determined whether the PDE is homogeneous or inhomogeneous, you have to figure out whether the PDE is linear or nonlinear.
A PDE of order $m$ is linear if and only if $u$ and all its derivatives appear linearly in the PDE and the coefficients of the terms only depend on the independent variables. Then a PDE of order $m$ is called semi-linear if the PDE is linear in the derivatives of the highest order ($m$) and the coefficients only depend on the independent variables. Finally, a PDE of order $m$ is called quasi-linear if it is linear in the derivatives of order $m$ with
coefficients that depend on the independent variables and derivatives of the dependent variable $u$ of order strictly less than m. In all other cases the PDE is called fully nonlinear.
Finally, some examples of PDEs are the transport equation, the wave equation and the heat equation. A more famous, still partially unsolved PDE are the Navier-Stokes equations.
Ordinary differential equations
The standard from of an ordinary differential equations is
\[ F(x^{(m)}, x^{(m – 1)}, …, x^\prime, x, t) = f(t). \]
These are difficult to solve in general. That is why I will only consider some types that are relatively easy to solve and go over the steps required to solve them. First the separable ODE.
\[ x^\prime = f(x) g(t)\]
Using the derivative a bit naively, we can separate the variables to find
\[ \frac{1}{f(x)} dx = g(t) dt.\]
Then by integrating both sides and solving for $x(t)$, we have found a solution to the differential equation. The second differential equation we consider is the linear ODE with constant coefficients. These are of the form
\[x^{(m)} + a_1 x^{(m -1)} + … + a_{m – 1} x^\prime + a_m x = f(t).\]
To solve these, the easiest way is to find the roots to the corresponding characteristic equation and then find one particular solution. The characteristic equation is given by
\[r^m + a_1 r^{m – 1} + … + a_{m-1} r + a_m = 0.\]
Let us only consider the case where all roots are distinct. In that case if we have roots $r_1, …, r_m$ then the homogeneous solution is given by
\[c_1 e^{r_1 t} + … + c_m e^{r_m t}. \]
Where $c_1,…,c_m$ are arbitrary constants. We then need to consider a particular solution. The easiest method to find a particular solution is to guess a solution that is similar to $f(t)$ and then plug this function into the differential equation and see if the solution works. This method is often referred to as Ansatz.
Partial differential equations
Partial differential equations are in general more difficult to solve than ordinary differential equations. The standard form of a PDE is given by
\[ F(\partial_{x_1}^m u,…,\partial_{x_n}^m u, \partial_{x_1}^{m – 1} u, …, \partial_{x_n}^{m – 1}, …, \partial_{x_i}^{a} \cdots \partial_{x_j}^{b} u, …, u, x_1, …, x_n) = f(x_1, …, x_n). \]
PDEs are very difficult to solve in general. We often need numerical methods to approximate solutions to PDEs. There are however a few famous types of PDEs with closed form solutions. These are already mentioned above. I will briefly go over the strategies to find the closed form solutions based on initial data.
The first equation is the transport equation. The standard form of the transport equation is given by
\[\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0.\]
Here we consider the initial data $u(0, x) = f(x)$. Then the solution to the transport equation is given by $u(t,x) = f(x – ct)$. To check this note that $u(0, x) = f(x – 0) = f(x)$. Furthermore, we have that $u_t(t, x) = f^\prime(x – ct) (- c)$ and $u_x(t, x) = f^\prime(x – ct)$. Here the subscripts mean the derivative with respect to the subscripted variable. Plugging these into the transport equation we see that we have found a solution.
The second equation we will consider is the one dimensional wave equation. The standard form of the wave equation is given by
\[\frac{\partial^2 u}{\partial t^2} – c^2 \frac{\partial^2 u}{\partial x^2} = 0.\]
Here we consider the initial data given by $u(0, x) = f(x)$ and $u_t(0, x) = g(x)$. The famous mathematician d’Alembert found a closed form solution to this problem. The solution is given by
\[u(t, x) = \frac{f(x – ct) + f(x + ct)}{2} + \frac{1}{2c}\int_{x – ct}^{x + ct} g(z) dz.\]
One can easily show that this solves the initial value problem.
The study into existence and uniqueness of solutions
In the study of differential equations, the first question one might ask is, if there even is a solution to a given differential equation. If a solution does indeed exist, a logical follow-up question becomes: given some initial data, is the solution to the differential equation unique or can we find multiple solutions. We often try to answer these questions even before even trying to solve the differential equation.
Emile Picard proved the existence of solutions to ODEs. He developed an iterative approximation technique that converges to a solution of a first order ODE based on some initial value. Since, the algorithm converges to a solution this proves that a solution to the initial value problem must exist. With this method Picard also managed to prove the uniqueness of the solution. The method to prove uniqueness of solutions is always the same. You start by assuming that there are two distinct solutions to a given differential equation. You then take the difference of these solutions and look at the implications that this has. If the solution is in fact unique given some initial data, you will find that the difference must be the zero function. This implies that the two distinct solutions are equal leading to a contradiction. This proves the uniqueness of the solution.
A nice example of such a proof is the proof of uniqueness to a solution of Poisson’s equation. This example involves the PDE given by Poisson’s equation, but the steps to this proof are very similar to the steps described above.
Conclusions
Differential equations are all around us and are sometimes difficult to understand. Finding solutions to differential equations always requires some tricks. Most often we first have to classify the differential equation correctly and can then solve it using a known set of steps. Seeing if a solution exists and determining the uniqueness of such solutions is also a very important part of differential equations. I hope to have created a roadmap for students that they can use to navigate the world of differential equations.