Introduction

In this article, we will introduce and motivate the idea of differential equations, which are simply equations containing differential terms $(dx, dy, dz)$.We shall introduce various terms used to describe the “character” of differential equations, such as order, degree etc. Throughout the article, we shall provide examples of differential equations, how to manipulate them, as well as a few solutions.

Equations vs. Expressions

At the elementary level, the difference between equations and expressions can be understood by stating that expressions lack an $=$ symbol, while equations are are statement of equivalence and hence contain the symbol. While the above explanation is certainly true, there exists are more nuanced method of differentiating between the 2: An expression provides arithmetic output, while an equation provides logical output.

Comprehension of the above statement is perhaps best achieved by illustration through a few examples. Consider the following expressions:
$$x^2 - 5$$ $$6x^5 - 7x^3$$ $$e^{5x} + \sqrt{\frac{\pi}{\sin{x}}}$$

In all of the above examples, plugging in any number/value in place of $x$ will result in the expression evaluating to another number/value. This form of numerical output is called arithmetic.

Equations, on the other hand, do not provide arithmetic output . For example, consider the following equations:
$$x^2 - 5 = 0$$ $$6x^5 - 7x^3 = 8x^2$$ $$e^{5x} + \sqrt{\frac{\pi}{\sin{x}}} = \frac{1}{\ln{x}}$$

Here, plugging in values for $x$ does not return a number. In the first example ($x^2 - 5 = 0$), try plugging in $x = 2$. On the left hand side, you get $2^2 -5$, which equals $-1$. On the right hand side, there is a $0$. Since both sides of the equation do not “equate”, the equation essentially returns the Boolean truth value FALSE. If we plug in $x = \sqrt{5}$, on the other hand, both sides of the equation become equal and the output will be TRUE. This form of TRUE/FALSE Boolean output is called logical.\ From this, it can be understood that expressions and equations are very similar in nature (essentially functions that take as input some quantity $x$), differing only in type of output (arithmetic for expressions, and logical for equations)

Differentials vs. Differential Operators

Another point of confusion typically encountered in the study of differential equations is the difference between differentials ($dx, dy, dz $), differential operators($\frac{d}{dx}$,$\frac{d}{dy}$,$\frac{d} {dz}$,$\frac{\partial}{\partial x}$), and derivatives ($\frac{dy} {dx},\frac{d^ny}{dx^n}, \frac{dz}{dx}, \frac{d^nz}{dx^n}$). We will provide clarity on these terms in this section before proceeding further.

A key point to note is that will not be using rigorous definitions, proofs, and theorems in this article, as our goal is mainly to understand, categorize, and solve differential equations. By this, I mean that occasionally, we perform operations such as “multiply both sides by $dx$”, which purists would argue are unwarranted. This is an extremely important point. While we can use shorthand “tricks” to quickly solve differential equations, one must consider the validity of such solutions. For example, how would you “multiply both sides by $dx$” if, instead of using the Leibniz notation $(\frac{d}{dx} f(x))$, I choose to use the operator notation ($D(f(x))$) or the prime notation $(f'(x))$ to denote the derivative of some function $f(x)$ ?

Such questions are of relevance for discussions on topics such as real analysis, where more rigorous definitions and proofs are required. Rest assured, all final conclusions obtained in this article are identical, equivalent, or at least valid in our context of discussion.

Differentials

Differentials, as you might have previously encountered in differential calculus, are simply infinitesimal changes in some variable. For example, $dx$ denotes an infinitely small change in $x$, and $dy$ denotes
an infinitely small change in $y$.

Derivatives

A derivative of some function $f(x)$ can be defined in the following way: $$ \frac{d}{{dx}}f\left( x \right) = \mathop {\lim }\limits_{\Delta \to 0} \frac{{f\left( {x + \Delta } \right) - f\left( x \right)}}{\Delta } $$ We assume that the reader is familiar with the usage and applications of the derivative.

There are many ways of conceptualizing the derivative. The first would be to think of it as a the ratio of 2 differentials. This arises from the usage of Leibniz notation. if we say that $f(x) = y$, then we can write the derivative of $f(x)$ as $\frac{dy}{dx}$, which looks almost like dividing $dy$ by $dx$. Note that this is not how we compute derivatives. $dy$ and $dx$ are not terms that we have fully defined (or written expressions for), so dividing them would prove to be an exercise in futility. Their ratio, however, is well defined (the derivative, as defined above). Nonetheless, we can treat $\frac{dy}{dx}$ as an algebraic expression for our purposes, as this line of thinking lends itself to methods of solving differential equations, which we will soon explore.

Differential operators

If, for example, we choose to think of the derivative not as a ratio of differentials, we can say that it is more akin to an operator, which acts on $f(x)$, ignoring any previous statement on infinitesimal changes, differentials, or ratios. In this view, the derivative would be the result of performing some operation on $f(x)$, which would be written as: $$\frac{d}{dx} (f(x))$$ Also, we might just as well choose some other notation in place of $\frac{d}{dx}$ to denote this operator, such as: $$D(f(x))$$

This form is called operator notation, and $D$,$\frac{d}{dx}$ are called differential operators.

It must be understood that there is no contradiction in viewing the derivative as a ratio of differentials or as the result of using the differential operator on a functions. These are simply 2 different perspectives on the derivative, which is computed the same way regardless of our choice of notation. The subtleties of the derivative is traditionally left a study in real analysis, and perhaps even linear algebra (where the derivative is considered a linear operator, and functions are taken to be vectors, which can result in tasks like finding eigenvalues and eigenvectors for the differential operator). For the most part, while discussing differential equations, we shall use the “ratio of 2 differentials” view.\ Note: In some cases, the differential operator is just taken to be $d(f(x))$. In this context, $d(f(x)) = f^\prime(x) dx$. However, in this article, we will continue to treat the “differential operator” as $\frac{d}{dx}$

A quick note on constants and variables

To quickly recap the differences between the types of scalar quantities used in differential equations, note the following:

Type Variable Value Known Value
Constant no yes
Arbitrary Constant no no
Variable yes no

As we shall soon see, one of the goals of writing differential equations
is eliminating arbitrary constants.

Order and degree

While considering polynomial expressions in some variable, say $x$, the highest power to which $x$ is raised to (the exponent) throughout the expression is of such relevance that is is given a name - degree. Here are a few examples to recap the idea: $$x^2 = 0 \quad\text{(degree 2)}$$ $$x^3 + 5x^2 = 0 \quad\text{(degree 3)}$$ $$x^7 + 8x^9 - 3x^{17} = 0 \quad\text{(degree 17)}$$ $$\sum_{i=0}^{n} {a_i x^i} = 0 \quad\text{(degree n)}$$

When we look at differential equations, however, there is also the order of the differential terms we need to consider. This characteristic (order) is the same one you might have encountered in differential calculus. Differential equations can also have degree. Here, it is defined to be the exponent of the term with highest order Again the notion of the order of a differential equation is best illustrated through a few examples: $$\left(\frac{dy}{dx}\right)^3 - 3\left(\frac{d^2y}{dx^2}\right)^2 +7y = 0$$ In the above equation, the highest order of any differential term is 2: $ - 3\left(\frac{d^2y} {dx^2}\right)^2$. Therefore, the order of the differential equation is 2. Next, you might think that the degree is 3, since it is the greatest exponent throughout the equation. However, as mentioned previously, the degree is found only after finding the order, and it is taken to be the greatest exponent of the term with the greatest order. Therefore, we first take the term with greatest order: $- 3\left(\frac{d^2y}{dx^2}\right)^2$, and find its exponent. Which, in this case, is 2. We can now conclude that we have a differential equation of order 2 and degree 2.

Let us consider another example: $$ \left( \frac{d^3y}{dx^3}\right)^3 - \left( \frac{dy}{dx} \right)^4 + \left( \frac{d^2y}{dx^2}\right)^2 - \left( \frac{d^4y}{dx^4} \right) = 0$$ Here, the order is 4, and the degree is 1. If we exclude the last term on the left hand side of the equation: $ - \left(\frac{d^4y}{dx^4}\right)$, the differential equation would be of order 2 and degree 3.

Differential equations can also contain radicals. To find the order and degree in this case, we need to exponentiate both sides of a radical equation until it is removed on both sides. Consider the following example: $$\sqrt{1-3- \left( \frac{dy}{dx} \right)^3} = \sqrt[3]{5+ 2\left( \frac{d^2y}{dx^2} \right)^2}$$ By raising both sides of the equation to the 6th power, we obtain: $$\left[ 1-3\left( \frac{d^3y}{dx^3} \right)^3 \right]^3 = \left[ 5+2\left( \frac{d^2y}{dx^2} \right)^2 \right]^2$$ From which we can conclude that the differential equation is of order 2 and degree 4 (by expansion of the expression on the right hand side).

Solutions to differential equations

To understand what the solution to a differential equation is, first, let us consider this simple equation: $$x^2 -1 = 0$$ The solutions to this equation, as it is traditionally referred to as, are simply the numerical values of $x$ that satisfy the above relation - $x= +1$ and $x= -1$. Here, the solutions to a simple equations are always numbers or simple expression, which can be evaluated to numerical values.

A differential equation, however, has a different type of solution- a function. That is, any function that satisfies the condition provided by a given differential equation is called a solution to that differential equation. For example, consider the following differential equation: $$\frac{d}{dx}f(x) = f(x)$$ Here, the solution to the differential equation would be the function $f(x) = e^x$, as it satisfies the condition above (you could plug in $f(x) = e^x$ into the differential equation and check if you like). The differential equation is essentially requesting for a function whose derivative is the function itself, and $e^x$ is a perfect candidate for the task. However, similar to how regular equations had multiple solutions, so can differential equations. If you notice, $f(x) = 0$ is an equally valid solution. Upon differentiation of the zero function, you are always returned the zero function.

Finding these solutions to non-trivial differential equations is usually far less straightforward than simply guessing the solution, like we did above. We shall explore a few elementary methods of finding solutions to more general differential equations shortly.

Forming differential equations

As it turns out, associated with a every family of (simple) equations is a differential equation. In other words, there exists some differential equation for which a given simple equation is a solution. To take an example, consider a standard parabola: $$y^2 = 4ax$$ Here, $a$ is an arbitrary constant called the focus of the parabola. $x$ and $y$ are variables. The presence of the arbitrary constant tells us that the above equation not only describes one parabola, but it defines an entire family of the. Constructing the differential equation for this parabola is as simple as repeatedly differentiating with respect to $x$, until the arbitrary constant is removed. $$2y\frac{dy}{dx} = 4a$$

Now, we are left with a choice. We could certainly differentiate again, as the right hand side of the equation is now a constant, and will vanish upon differentiation. The evaluation of the left hand side, however, would require the usage of the product rule, which is an unnecessary complication for this simple problem. A more clever approach would be to eliminating $a$ by plugging in its definition from the second equation back into the first. $$4a = 2y\frac{dy}{dx}$$ $$y^2 = 4a x$$ $$y^2 = 2y\frac{dy}{dx} x$$ $$y = 2x\frac{dy}{dx}$$ The last equation shown in the list above is the final, simplified differential equation for the family of parabolas of the form $y^2 = 4ax$. As you might have noticed, the final differential equation that we obtained ($ y = 2x\frac{dy}{dx}$) is free from arbitrary constants.

This is actually one of the primary purposes of writing differential equations- to have a general equation that is free from arbitrary constants to describe a family of curves. This is an extremely important and relevant point to take note of.

**Theorem: ** The differential equation of a source curve with $n$ arbitrary
constants will be of order $n$

The proof for the above theorem is beyond the scope of this article. Let us now test the above rule with a few examples. To start off with, let us find the differential equation of a straight line on the $x-y$ plane. The source equation is: $$y = mx+b$$ Where $m$ and $b$ are arbitrary constants that represent the slope and $y$-intercept respectively. According to the theorem, this equation must have a corresponding second-order differential equation with no arbitrary constants. To obtain this, we can simple differentiate twice: $$\frac{dy}{dx} = m$$ $$\frac{d^2y}{dx^2} =0$$

Just as predicted, the differential equation that involves no arbitrary constants is of order 2. Next let’s apply a similar method for another equation: $$y = e^{ax}$$ By differentiating, $$\frac{dy}{dx} = ae^{ax} = ay$$

Multiplying both sides by $x$, $$x\frac{dy}{dx} = xay$$ Since $ax = \ln{y}$, $$x\frac{dy}{dx} = y\ln{y}$$ Which is our final differential equation for $y = e^{ax}$.
Again, the theorem holds true. Initially, we had just one arbitrary variable, $a$. As a result, the resulting differential equation was of the first order.

There are many more useful techniques that one can use to find the associated differential equation for a source curve. In general, the difficulty rises as the number of arbitrary constants in the source curve increases. While not a rigorous rule, you are generally expected to differentiate $n$ times to obtain the differential equation associated with a source curve with $n$ arbitrary constants. We leave the task of practising this technique to the reader.