What Is a Differential Equation?

A differential equation is simply any equation containing one or more derivatives of any kind.

$\frac{d}{dx}\sin(x) = \cos(x)$

is a differential equation because it contains a derivative on the left-hand side. However, perhaps it will be written as

$\frac{d\sin(x)}{dx} = \cos(x)$

$\frac{dy}{dx} = e^x + 5$

is also a differential equation; the derivative is just written in a different form, the derivative of $y$ with respect to $x$.

Differential equations can be solved to find the function that is differentiated. This typically involves integration (antidifferentiation) combined with a variety of other techniques, but there are a overwhelming number of different ways to solve different types of differential equations.

A differential equation is an equation of a function which provides a way to find its derivative. In the real world this usually corresponds to a rate of change.

However, sometimes a function has multiple variables. In such a circumstance you must differentiate the function with respect to one variable at a time. For example, if a function's variables are x and y, you would first differentiate it with respect to x while treating y as any other constant, and then do the exact opposite. This is called a partial derivative. ie: dz/dx[(x^2)+4y]=2x dz/dy[(x^2)+4y]=4

Sometimes you will get a derivative of zero when an equation has multiple variables. Such an instant is called a saddle point for the shape the graph makes at that distinct instant, it is simultaneously a maximum and a minimum, but of different variables respectively.

A Differential Equation is just the equation of the graph of the slopes of every point in a given graph. Let the given graph be a function of x say f(x). The differential equation represented by $\frac{df\left(x\right)}{dx}$ gives us the slope of the graph at a given point.

The above graph is that of function $x^{​2}$+2 When we differentiate the function we get 2x (I'll tell you how I got that in the next section). This is the graph of the slope of the function