In many cases, if you would like to find out how many unknown functions there are on a function, then you can just plug in the values for all the unknown functions one by one into the equations of the partial differential equation, but this may not always produce the desired output. In cases where the output is not a constant, then you will need to plug in the values for the unknown functions one at a time, in order to get the results you are looking for. For example, if you have an unknown constant such as x, then you could plug in your results as the slopes of the x-axis.
Differential equations can be used to solve for the unknown function f(x), or to find the function of a specific integral. In the first example, you can plug in whatever value of x you would like to solve for, such as “tan”, and then you just need to plug in your result into a partial differential equation such as the tangent function, or the intercept of a set of exponential curves, etc. In the second example, you would plug in “sin(x)”. This will give you the sine of the tangent curve, which will then determine the intercept. In either example, if the slope of the tangent function (or the intercept) occurs at the exact location when the function is graphed out, then you know that the function exists for that given input value of x.
The more common uses of differential equations are in optimization. You can easily solve for the parameters of a dynamic function using a differential equation. Say for instance that we want to optimize a point function, which takes a point as an input, and returns the maximum value of the function when that point reaches a certain value (for example, the slope of a line). Using the partial derivative of the function, we can quickly solve for the parameters of the function, without having to perform any real-time optimization algorithm.
Another application of differential equations is when dealing with the time domain. When fitting a model to a set of data, such as in an economic model, the model often must be fit using stable differential equations. That is, if you were to take the normal derivative of the function with respect to time, and allow it to vary, it would not yield the expected result. By fitting the model to the data using a differential equation, we can ensure that the function is linearly fit to the data. Thus, it becomes much easier to implement and measure economic models over time.
Of course, differential equations can also be used in other areas as well. Here we will discuss three examples, all of which are important to the development of mechanical design. We will discuss differential forms for functions which take exponential or logarithmic values, a function that changes only during a fixed time interval, and a tangent function, which, as the name indicates, tangents multiple derivatives. All these topics, and many others, are touched on in greater depth in greater detail in a series of further articles.
In any case, one can quickly memorize the differential form for the integral function as illustrated above. Then by using the differential equation, we can find the integral curve, as illustrated in the plot below. This shows the change in the integral value, plotted against time, as the integral curve varies linearly along a definite path. Other differential equations that are widely used in the design of engines, turbines, airplanes, rocket motors, missiles, and space vehicles include those relating the differential operator function, which transforms a polynomial expression to a differential, as well as the differential operator and integral operator functions. The third example relates the differential tangent function to the differential derivative of a function. Here the dotted line represents the partial derivatives of the function, when the function is plotted against time.
The articles listed here serve a single purpose: to introduce differential equations for a specific application. In order to learn more about differential equations in general, a future article will address that topic. For now, however, this brief introduction serves to introduce you to a few of the essential differential equations you may encounter in your studies of numerous fields. You can quickly memorize them as illustrated above, but much of the understanding rests upon further learning about the properties of each equation. Studying more advanced mathematics will greatly enhance your comprehension of differential equations in future articles.