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An equation containing derivatives or differentials with one or more dependent variables
Ordinary differential equation
An ordinary d.e. contains only ordinary derivatives or differentials of one or more dependent variables with respect to a single independent variable
Partial differential equation
An equation involving partial derivatives of one or more dependent variable(s) with respect to two or more independent variables
Order of a differential equation
the order of a d.e. is the highest order derivative in the d.e.
Linear or Nonlinear differential equations
In a linear 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
A solution of a differential equation
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 _____ at a, it must be ______ at a
If a function is differentiable at a, it must be continuous at a.
But the converse is NOT TRUE.
Part I of the Fundamental Theory of Calculus
If f(t) is a continuous function on [a,b], define
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.
An implicit solution of a d.e.
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.
Particular solution to a d.e.
A solution to the given d.e. is called a particular solution if it does not contain any parameters in it.
Higher Order Partial Derivatives
- If fx(x,y) and fy(x,y) are differentiable functions, then
Differential (Total Differential)
- Let z=f(x,y) be a differential function then
- dz=fx(x,y)dx+fy(x,y)dy is the total differential of z=f(x,y)
An exact DE
- 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
Separable in Variable
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