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Conventions

Lesson 2:

Variables and Parameters

In genereal, four types of quantities can appear in equations:

Constant

A fixed number such as a mathematical constant, like π or e which can never change, a physical constant, which is defined or experimentally determined, or a given constant for a situation such as the width of a gap to be bridged.

Parameter

A parameter occurs in an equation to indicate that the equation actually stands for a family of equations which differ only in the value of the parameter. The so-called constant of integration is a pertinent example.

Dependent and Independent Variable

The distinction between dependent and independent variables is often purely arbitrary in algebra problems. {$ y = 3x + 5 $} can just as well be {$ x = (y-5)/3 $}. In more practical problems, one or more variables are regarded as independent. These are quantities which may be given or found by experiment. On the other hand, one variable is the dependent variable. It is determined from whatever values are found for the independent variables. Generally, the independent variable can be expressed as a function of the independent variables, such as:

{$$ y = f(x) = x^2 + 3x + 1 $$}

in which y is the dependent variable, or a function of the independent variable x. However, there are also equations which are implicit functions in which it is impossible to separate the variables.

The Exponential Function

The exponential function is a function in which a number is raised to a variable power. Any positive number raised to a variable power would be a exponential function, but we are almost exclusively concerned with the number e, which when raised to a variable power is the variable function.

Exponential functions should not be confused with functions in which a variable is raised to a numerical exponent. In exponential functions, a variable is in the exponent.

As this subject involves exponents, reviewing the laws of exponents is in order: Laws of Exponents Review Sheet. Especially to be noted is that since e is a positive number, the exponential function can never be negative and no value of the exponent can ever make it zero.


More about exponential functions at Exponential and Logarithmic Functions

Derivative Notation

There are four major types of notation for derivatives.

{$ \displaystyle \Large{ dy\over{dx} }$}

This is Leibniz's notation. It is very popular because it specifies the independent variable. It is echoed in notation for integrals. The y is sometimes replaced by f(x) or is omitted altogether when the whole function follows. Successive derivatives are indicated by

{$$ {d^2y\over{dx^2}},\ {d^3y\over{dx^3}},\ {d^4y\over{dx^4}},\ \text{etc.} $$}

{$ \displaystyle \Large{ f^\prime (x) \text{ or } y^\prime }$}

Prime notation was used by Newton and Lagrange. Successive derivatives are indicated with more primes.

{$ \displaystyle \Large{ Dy }$}

This is Euler's notation. Exponents on the D indicate successive derivatives.

This notation is also used for the differential operator.

{$ \displaystyle \Large{ \dot{x} }$}

The dot notation was used by Newton and is most often used for derivatives with respect to time. Additional dots represent successive derivatives.

Differential Equations

Differential equations (DEs) express a relation between a function and its derivatives.

Identifying Differential Equations

A DE is an ordinary differential equation (ODE) if the function involved has only one independent variable.

A 'first order' DE involves only the first derivative.

Verifying Solutions

Proposed solutions to a DE can be tested simply by substituting the proposed solution in the DE to see whether the DE holds.

Initial Value Problems (IVP)


<< First Order OEDs | Trail MIT 18.03SC Differential Equations | Basic DES >>

Sources:

Edwards, C., and D. Penney. Elementary Differential Equations with Boundary Value Problems. 6th ed. Upper Saddle River, NJ: Prentice Hall, 2003 [EP]

Recommended:

  • [EP]: 1.1 and 1.4

Categories:

Tags: Xxx First Order Differential Equations , Xxx Ordinary Differential Equations

Includes:


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