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Differential Equations (DE) - Ordinary Differential Equations (ODEs) - Partial Differential Equations (PDEs)

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Differential Equation (DE)
  • is an equation that relates one or more functions and their derivatives
  • is an equation with a function and one or more of its derivatives
  • Ordinary Differential Equations (ODEs) - deal with derivatives of a function with respect to a single independent variable
  • Partial Differential Equations (PDEs) - involve partial derivatives of a function with respect to multiple independent variables

DE - Tutorials

DE - Intuition

Say the population increases 𝑦(𝑡)’ as the growth rate 𝑟 times the population 𝑦(𝑡) at time instant 𝑡. This states a differential equation shown below:

Indent

𝑦(𝑡)’ = 𝑟·𝑦(𝑡)

The problem is to find an equation 𝑦(𝑡) or set of equations that satisfy the differential equation above.

The solution is a class of functions of the form:

Indent

𝑦(𝑡) = 𝑎·𝑒𝑟𝑡# where 𝑎 can be any scalar value

The differential equation above is of type First-Order Linear Constant Coefficient Homogenous Differential Equation.
There are several other types as shown below.
But there is no magic bullet to solve all Differential Equations

DE - Types

dimensions:

  • order (1st-Order, 2nd-Order, …, Nth-Order)
  • linearity (linear or non-linear) - a differential equation is linear if the equation is a linear combination of {𝑦, 𝑦’, …, 𝑦(𝑛)}
  • separability (separable or non-separable) - a differential equation is separable the following 2 conditions are met:
    • all the (𝑑𝑦 & 𝑦)‘s can be moved to one side of the differential equation
    • all the (𝑑𝑥 & 𝑥)‘s can be moved to another side of the differential equation

Order

Form

Specific Forms

First-Order DE

𝑦’ = 𝐹(𝑦, 𝑥)

Second-Order DE

𝑦” = 𝐹(𝑦’, 𝑦, 𝑥)

Nth-Order DE

𝑦(𝑛) = 𝐹(𝑦(𝑛-1), …, 𝑦, 𝑥)

DE - Methods to Solve Differential Equations

Below is fromPhysics Students Need to Know These 5 Methods for Differential Equations

Method

Description

Substitution

TODO from the link above

Energy Conservation

TODO from the link above

Series Expansion

TODO from the link above

Laplace Transform

  • is a mathematical method that converts a function of a real variable to a function of a complex variable
  • a function 𝑓(𝑡) that can’t be Fourier Transformed can be Laplace Transformed by multiplying the function by a decaying exponential 𝑒-𝜁𝑡 and a heavyside function 𝐻(𝑡) where 𝜁 is a constant:
    • 𝐹(𝑡) = 𝑓(𝑡)𝑒-𝜁𝑡𝐻(𝑡)
    • thus:
      • the Laplace transform of 𝑓(𝑡) is the Fourier transform of 𝐹(𝑡)
      • the Laplace transform is a one-sided weight Fourier transform
  • a discrete version is Z-Transform
Link to original

Hamiltonian Flow

TODO from the link above

DE - Subpages