Differential Equations

Todo:

Map out method (order of operations) to solve a differential equation
This should start by classifying the differential equation then guiding the user to the best method to solve the equation
Reorganize the notes about the different types of differential equations
Reorganize the notes about the solving methods
Reorganize the loose definitions and other notes
Makes graphics and clean up explanations
Python section (source material)

At its most broadest definition differential equations are equations that contain a derivative in some capacity. In application, these equations generally take the from:

Where , , and are all function of t and is the derivative of . As you can see the equation has the function and its derivative , this can lead some interesting properties as the equation can be rearranged to find that is a function of it's own anti-derivative.

They can be used to model complex systems with multiple interdependent rates of change. For example the SIR Model often used to simulate the transmission of a disease through a population.

Solving

The goal when solving a differential equation is to find the general solution, or the function that satisfices the original equation. Due to the complex nature of differential equations not all methods work on all problems. In order to solve our equation we must first classify our equation:

Is it a scalar equation (one equation) or a systems of equations?
What is the highest derivative order within our equation(s)
Is the equation linear or non-linear?

See the Classifying Differential Equations section for more details on this process.

Once we have our equation(s) classified we can this look through our methods to solve them:

If you are trying to solve a scalar equation:

Integrating Factor if your equation is:
- linear
- 1st order.
Variation of Parameters if your equation is:
- linear
- 1st order
Undetermined Coefficients if your equations is:
- to be expanded
Separation of Variables if your equation is:
- linear or nonlinear
- any order

If you are trying to solve a system of differential equations:

Due to the complex nature of this problem there are many ways to approach and subsequently solve the equation. The method you choose depends on the differential equation or system of differential equations in question.

If you problem provides the output of the equation at some starting point the problem is then called an initial value problem.

Classifying Differential Equations

Differential Equations are classified using using 2 factors:

The highest order of the derivative used in the equation.
Whether or not the equation is linear.

Linear Differential Equations

Linear Differential Equations follow the form:

if a differential equation cannot be put into this form it is said to be non-linear.

complete and link first diffeq notes