D.E. Definitions

An equation containing derivatives or differentials with one or more dependent variables

An ordinary d.e. contains only ordinary derivatives or differentials of one or more dependent variables with respect to a single independent variable

An equation involving partial derivatives of one or more dependent variable(s) with respect to two or more independent variables

the order of a d.e. is the highest order derivative in the d.e.

• a d.e. or order n is said to be linear if it has the form:
• ...

otherwise it is called an nonlinear d.e.

• a) The dependent variable y and its derivatives y,y',y'',...,yⁿ are of the first degree, that is, the order of each is one.
• b) The coefficients of y,y',y'',...,yⁿ depend at most on independent variable x

is a function f defined on some interval I, which when substituted into the d.e. reduces the equation to an identity

If a function is differentiable at a, it must be continuous at a.

But the converse is NOT TRUE.

If f(t) is a continuous function on [a,b], define , then

Given an equation relating x and y or a relation G(x,y)=0, if we can solve y in terms of x, then y=f(x) is called an implicit function defined by the given equation, that satisfies the equation.

A relation G(x,y)=0 is an implicit solution to the given d.e. if there exists an implicit function y=f(x) defined on some interval I which satisfies the relation G(x,y)=0 as well as the d.e. on I.

A solution to the given d.e. is called a particular solution if it does not contain any parameters in it.

• If f_x(x,y) and f_y(x,y) are differentiable functions, then
• 
• 

• Let z=f(x,y) be a differential function then
• dz=f_x(x,y)dx+f_y(x,y)dy is the total differential of z=f(x,y)

• A fist order linear or nonlinear d.e. of the form
• M(x,y)dx+N(x,y)dy=0 or M(x,y)+N(x,y)dy/dx=0 is said to be an exact d.e. if there exists a function f(x,y) defined on some region R of the xy-plane such that

• The function μ(x) or μ(y) that when multiplied with a non exact d.e. M(x,y)dx+N(x,y)dy=0 converts it into an exact d.e.
• The resulting d.e. may not be equivalent to the original but the solution of one is also the solution of the other.

A function f(x,y) is said to be a homogeneous function of degree n≥0∈ℝ if f(tx,ty)=tⁿ[f(x,y)] for a non-zero constant t or t≠0

A d.e. of the form is said to be a homogeneous d.e. if M(x,y) and N(x,y) are homogeneous functions of the same degree n≥0∈ℝ

• A differential equation of the form
• where y≠0 and n∈ℝ

A first order in the form . If f(x,y) can be expressed as a product of g(x) and h(y), then is separable in variable, or