 Let's look at Hooke's law as a simple example. F is equal to minus D times y. This law describes the restoring force F on a mass attached to a spring. The mass experiences this force when you displace it by the distance y from the equilibrium position. D is a constant coefficient that describes how hard it is to stretch or compress the spring. The mass M of the ball attached to the spring is hidden in the force. We can write the force according to Newton's third law as M times A. A is the acceleration that the mass experiences when it is displaced by the distance y from its rest position. As soon as you pull on the mass and release it, the spring will start swinging back and forth. But friction is in this case, it will never stop swinging. While the mass oscillates, the displacement y changes. The displacement is therefore dependent on the time t. Thus also the acceleration A depends on the time t. The mass of course remains the same at any time, no matter how much the spring is displaced. This is also true in good approximation for the spring constant D. If we now bring M to the other side, we can use this equation to calculate the acceleration experienced by the mass at each displacement y. But what if we are interested in the question at which displacement y will the spring be after 24 seconds? To be able to answer such a future question, we must know how exactly y depends on the time t. We only know that y does depend on time, but not how. And exactly when dealing with such future questions, differential equations come into play. We can easily show that the acceleration A is the second time derivative of the distance traveled. So in our case it is the second derivative of y with respect to time t. Now we have set up a differential equation for the displacement y. You can recognize a differential equation in short dq by the fact that in addition to the searched function y, it also contains derivatives of this function. Like in this case the second derivative of y with respect to time t. A differential equation is an equation containing a searched function y and derivatives of this function. You will certainly encounter many notations of a differential equation. We have written down our differential equation in the so called Leibniz notation. You will often encounter this notation in physics. We can also write it down a bit more compactly without mentioning the time dependence. If the function y depends only on the time t, then we can write down the time derivative even more compactly with the so called Newton notation. One time derivative of y corresponds to one point above y, so if there is a second time derivative as in our case, there will be two points. Obviously, this notation is rather unsuitable if you want to consider the tenth derivative for example. Another notation you are more likely to encounter in mathematics is the Lagrange notation. Here we use primes for the derivatives. So for the second derivative, two primes. In Lagrange notation it should be clear from the context with respect to which variable the function is differentiated. If it is not clear, then you should write out explicitly on which variables y depends. Each notation has its advantages and disadvantages. However, remember that these are just different ways of writing down the same physics. Even rearranging and renaming does not change the physics under the hood of this differential equation. To answer our previous question at which displacement y will the spring be after 24 seconds, we must solve the post differential equation. Solving a differential equation means that you have to find out how the function y you are looking for exactly depends on the variable t. For simple differential equations like the one of the oscillating mass, there are two solving methods you can use to find the function y. Keep in mind however that there is no general recipe for how you can solve an arbitrary differential equation. For some differential equations there is not even an analytic solution. Here the word analytic means that you cannot write down a concrete equation for the function y. The only possibility in this case is to solve the differential equation on the computer numerically. Then the computer does not spit out a concrete formula but data points which you can represent in a diagram and then analyze the behavior of the differential equation. Once you encounter a differential equation, the first thing you need to figure out is which one is the function you are looking for and which variables it depends on. In our differential equation of the oscillating mass, the function we are looking for is called y and it depends on the variable t. As another example, look at the wave function that describes the electric field of an electromagnetic wave propagating at the speed of light c. What is the function you are looking for in this differential equation? It is the function e because its derivatives occur here. And which variables does the function e depend? The dependence is not explicitly given here but from the derivatives you can immediately see that e must depend on x, y, z and on t. That is on a total of four variables. Let's look at a slightly more complex example. This system of differential equations describes how a mass moves in a three-dimensional gravitational field. Here you have a so-called coupled differential equation system. In this case, a single differential equation is not sufficient to describe the motion of a mass in the gravitational field. In fact, three functions are searched here, namely the trajectories x, y and z, which determine a position of the mass in three-dimensional space. Each function describes the motion in one of the three spatial directions. And all three trajectories depend only on the time t. What does it even mean if we have coupled differential equations? The word coupled means that, for example, in the first differential equation for the function x there is also the function y. So we cannot simply solve the first differential equation independently of the second one because the second equation tells us how y behaves in the first equation. In all three differential equations, all of the searched function x, y and z occur, which means that we have to solve all three differential equations simultaneously.