8 - Recap on The Fundamentals of Differential Equations

Instead of writing them as , assuming variables y and x, we will write them as:

This is the partial derivative of y as a function of x. The partial derivative is the same process as taking the typical derivative, but instead of performing implicit differentiation on other variables, we just assume any other variables as constants.

For example, if the derivative of, say, 3x+y would be:
The partial derivative would be:

Cauchy problems are just differential equations where you have a starting condition, a specific value of f(x) at some point, or something similar. You solve this like a normal differential equation to get your general solution.

In the case of a Cauchy Problem, we have a specific value we want for f(x), so this general solution cannot apply anymore.

This is where you use the particular solution.

The particular solution is the general solution, but instead of a constant, you will have to find that specific value of c for which this solution is valid.

They are differential equations that you have to solve by first just separating variables, bringing all the x and dx on one side and the y and dy on the other before integrating both sides. That's it.

Linear differential equations are the case when you can't simply separate, as there would be a multiplication or addition that doesn't allow us.

Where is the most important part. It's a polynomial or function of x. is the rest of the equation, dependent on x. It's important to have that derivative alone. If you find you have something multiplying or dividing that, multiply the entire equation or divide it or do whatever you can to get it alone. That is the result of a product rule, except that it's missing something.

The goal is to complete the product rule missing.
You will then use to calculate the rho, the integrating factor: You then multiply the entire equation by this , and you end up with the complete product rule, which is going to, in itself, be a derivative. You can now easily integrate both sides knowing that's the derivative of something.