Classifying Differential Equations

Differential equations (DEs) come in two basic forms

Ordinary differential equations (ODEs) — those that involve the relation between two variables only.
Partial differential equations (PDEs) — those that involve more than two variables and hence partial derivatives. These are generally trickier. Importantly, the presence of a $𝜕$ symbol does not imply a DE is a PDE, for if there are only two variables present it is an ODE.

Other attributes

Order

Both types of DE have order, that is the highest order of derivative (mixed or unmixed) in the equation.

$𝑑 2 𝑦 𝑑 𝑥 2 - 2 = 𝑦$ is order 2.
$𝜕 5 𝑓 𝜕 𝑥 3 𝜕 𝑡 2 + 𝜕 2 𝑓 𝜕 𝑥 2 + 𝜕 𝑓 𝜕 𝑡 = 0$ is order 5.

Degree

The degree of a DE is the highest power of the highest derivative. An example of a degree 2 DE is the DE for the Brachistochrone problem.

[1 + (𝑑 𝑦 𝑑 𝑦) 2] 𝑦 = 𝑘 2

Linearity

Both can also be either linear or non-linear. For a DE to be linear means it is a linear combination of derivatives (of any order), i.e. there are no products of derivatives.

Homogeneousness

A second order linear differential equation can be homogeneous or non-homogeneous. In general, they have the form

𝑑 2 𝑦 𝑑 𝑥 2 + 𝑝 (𝑥) 𝑑 𝑦 𝑑 𝑥 + 𝑞 (𝑥) = 𝑔 (𝑥)

If $𝑔 (𝑥) = 0$ , the equation is homogeneous.

tidy | SemBr

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