A differential equation is called separable if the dependent and independent variable can be separated to different sides of the equation. This allows us to (potentially) solve this sort of differential equation by elementary integration techniques. We'll consider some examples. We'll also explore solving a differential equation precisely by means of what is called an initial condition. It is called an initial condition because it generally corresponds to a value that might be experimentally known for the functional solution. Note, in the solution we will see some manipulation of constants. In general, we will note simply that a sum, product, or exponent raised to a constant is again a constant and we will not be particularly careful to note the differences between those constants.
To the right right, we see what is referred to as a slope-field diagram with the specific solution to this differential equation overlayed in black. To generate the slope field, I choose values for y and t and that yields a value for y' which is the derivative (or slope) at that particular point. I then graph a small line segment at the point with that slope. Solutions are generally traces through the slope field, but this methodology is not terribly helpful for differential equations with higher order derivative terms like y'' or y'''. Additionally, it may simply be misleading or too difficult for the human eye to interpret the various families of solutions to be found on the slope field. Try a second example below.
Python Programming Project
Write a program that creates a slope field of a given differential equation with the structure y' = f(t, y) where t and y may appear in any reasonable combination. Consider the examples
y' = t**2 + y
y' = t*y+t
y' = t**2
y' = y + t
y' = y*exp(t)
The examples are given in standard python notation where ** denotes that the following number will be an exponent and * denotes multiplication. Notice that you can solve some of these with the technique of separation, but not all. Oftentimes, choosing different regimes (+, 0, -, very small, very large, ect.) for the given constants will give you a way to determine what solutions might look like. Over time you will develop a good intuition for this as well.