 separation of variables. With this method you can solve ordinary differential equations of first order, which must also be linear and homogeneous. This type of differential equations has the form y' plus k times y is equal to 0. The coefficient k must not necessarily be constant but can also depend on x. Also note that before the first derivative y' the coefficient must be equal to 1. If this is not the case then you simply have to divide the whole equation by the coefficient which is in front of y'. Then you have the right form. In this solving method y and x are considered as two variables and separated from each other by bringing y to one side and x to the other side of the equation. The Leibniz notation of the differential equation is best suited for this purpose. Bring k times y to the right-hand side. Multiply the equation by dx and then divide the equation by y. This way you have only y dependence on the left side and only x dependence on the right side. Now you can integrate over y on the left-hand side and over x on the right-hand side. Integrating 1 over y gives the natural logarithm of y. Also don't forget the integration constant. Let's call it a for example. Now you have to solve for the function y. Use the exponential function on both sides. You can split the sum in the exponential term on the left side into a product where e to the power of ln of y is simply y. Bring the constant e to the power of a to the right side and rename the coefficient to a new constant c. As a result you get a general solution formula that you can always use to solve homogeneous differential equations. You don't have to apply the separation of variables method again and again, but you can use the solution formula directly. For example let's look at the differential equation for the radioactive decay law. In this case the search function y is the number of not yet decayed atomic nuclei n and the variable x corresponds to the time t. And the coefficient k in this case is a decay constant lambda. According to the solution formula you have to integrate the coefficient that is the decay constant lambda over t. Integrating a constant just gives t. And we already have the general solution for the decay law. Now you know the qualitative behavior of the physical process namely that atomic nuclei decay exponentially. But you cannot say concretely yet how many nuclei have already decayed after a certain period of time. This is because you don't know the constant c yet. After all in the decay law c gives the number of atomic nuclei that were present at the beginning of your observation. So you need an initial condition as additional information to the differential equation. It could be for example like this n of 0 is equal to 1000. That means at the time t equals 0 there were 1000 atomic nuclei. Inserting this initial condition results in this equation. e to the power of 0 is 1 so c must be 1000. Now you can insert an arbitrary point of time and find out how many not yet decayed atomic nuclei are still there.