Differential Equations (DEs)

Classifying DEs

Definition 1: Independent and Dependent Variables and Parameters

The dependent variable(s) of a DE are the unknown functions that we want to solve for i.e. $f(x),y(x,t),$ etc.

The independent variable(s) of a DE are the variable(s) that the independent variable(s) depend on i.e. $x,t,$ etc.

A parameter is a term that is an unknown but is not an independent or dependent variable i.e $a,b,\alpha ,\beta ,$ etc.

Definition 2: Order of a DE

The order of a DE is the order of the highest derivative.

Definition 3: ODEs and PDEs

A DE is an ordinary differential equation (ODE) if it only contains ordinary derivatives (i.e. no partial derivatives)

A DE is a partial differential equation (PDE) if it contains at least one partial derivative of a independent variable.

Definition 4: Linear and nonlinear DEs

A DE that contains no products of terms involving the dependent variable(s) is called linear.

If a DE is not linear then it is nonlinear.

Example:

Definition 5: Homogeneous and Inhomogeneous: DEs

DE where every term depends on a dependent variable is called homogeneous.

A DE that is not homogeneous is called inhomogeneous or nonhomogeneous.

Linear homogeneous DEs have the property that if $f_1$ and $f_2$ both solve the DE then so does $a{f}_1+b{f}_2$ for all$a,b\in \mathbb{R}$.

This is the same property that was used to define linearity in MATH115! i.e. a vector valued function $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is linear if and only if for all $\vec{x},\vec{y}\in {\mathbb{R}}^n$ and for all $a,b,\in \mathbb{R},f(a\vec{x}+b\vec{y})=af(\vec{x})+bf(\vec{y})$.

The difference is that we now study linear functions applied to the vector space of all sufficiently differentiable functions (e.g. $C^{\infty }(\mathbb{R})$) instead of vectors in ${\mathbb{R}}^n$ i.e. there are no matrix representations for linear functions.