Differential equations are divided into 2 large categories – ordinary differential equations or ODES involving functions with one variable, most often in the face of time and partial differential equations with several variables. If partial differential equations describe more complex characteristics, for example, temperature changes at different points in space, then ordinary differential equations describe more static characteristics that change over time.

As a good example, we can consider the process of falling of an object. As you know, the gravitational acceleration is 9.81 m/s2, which means that if you analyze the position of the body at every second and translate this state into vectors, they will accumulate an additional downward acceleration of 9.81 m/s2 every second. This gives an example of the simplest differential equation, the solution of which is the function y (t), the derivative of which gives the vertical component, and the velocity gives the vertical component of acceleration (1).

This equation can be solved by allocating (2) for speed and (3) for path.

Another interesting moment is when it is possible to describe the movement of celestial objects on this scale due to the force of gravity. So, two bodies are given whose attraction is directed towards each other with a force inversely proportional to the square of the distance between them (4).

It is known that the derivative of the coordinate is velocity, the derivative of velocity is acceleration and it is necessary to obtain a function for motion, but according to equation (4), only the equation for acceleration (5) is known.

It may be strange here that the derivative is equal to the same function, but this is a common phenomenon when the derivative of the first or higher orders is determined by the values of themselves. But in practice, it is more often necessary to work with second-order differential equations, as can be seen in the previous examples.

However, there are also differential equations with third (6) or fourth (7) derivatives or higher (8) derivatives, which are considered higher-order differential equations.

In a way, it turns out that you need to find infinitely many numbers, one for each moment of time, but in general this coincides with the description of the function. And most often, even if in many cases it is possible to apply the classical description, then to a greater extent the use of the technology of ordinary mathematical transformations no longer meets the requirements. The usual description of the characteristics of a mathematical pendulum can be a proof of this.

Considering the real and idealized case, it can be noted that idealization works only at small angles of deflection of the pendulum, but when the angle becomes large enough, for example, equal to a semicircle, then the graph describing its oscillations as a whole ceases to be similar to the graphs of sine or cosine. The reason for this is the need to describe its motion exclusively using not partial, but general equations of harmonic oscillations with second-order differential equations.

This analogy can be applied to many other physical, most often real phenomena.

1. Pontryagin L. S. Ordinary differential equations. – M.: Nauka, 1974.

2. Tikhonov A. N., Samarsky A. A. Equations of mathematical physics. – M.: Nauka, 1972.

3. Tikhonov A. N., Vasilyeva A. B., Sveshnikov A. G. Differential equations. – 4th ed. – Fzimatlit, 2005.

4. Umnov A. E., Umnov E. A. Fundamentals of the theory of differential equations. – Ed. 2nd – 2007. – 240 p.