1. General comprehension questions:

1. What is a differential and what forms of differentials do you know?

2. What does it mean for a differential equation to be exact?

A list of words you might need for the answer:

A function of x and y; …times y prime; whose derivative with respect to x; a derivative of…; we take a partial derivative of… with respect to…; 2xe to the y; sin; cos; integrate with respect to…; a generic function; a condition; the final solution; an arbitrary constant, etc.

2. Learn to read the following formulas:

1) P(x,y)dx + Q(x,y)dy:

P of x and y multiplied by dx plus Q of x and y multiplied by dy;

2)

df is equal to the partial derivative of f with respect to x multiplied by dx plus the partial derivative of f with respect to y multiplied by dy;

3)

Second partial derivative of f with respect to x and y;

4) y=∫f(x)dx+c

y is equal to the integral of f of x with respect to x plus constant;

5) y=ce^-kx

y is equal to c times e to the minus k x

Read Text 1 aloud with the formulas:

Text 1.

A differential of the form

(1)

is exact (also called a total differential) if is path-independent. This will be true if

(2)

so and must be of the form

(3)

But

	(4)
	(5)

so

(6)

There is a special notation encountered especially often in statistical thermodynamics. Consider an exact differential

(7)

Then the notation , sometimes referred to as constrained variable notation, means "the partial derivative of with respect to with held constant." Extending this notation a bit leads to the identity

(8)

where it is understood that on the last-hand side is treated as a variable that can itself be held constant.

*Source: Weisstein, Eric W. Exact Differential. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ExactDifferential.html

Vocabulary

1. A list of words to remember:

application – применение, приложение

arbitrary - произвольный

be referred to as - называться

to derive – выводить

derivative – производная

partial derivative – частная производная

differentiate with respect to… - дифференцировать по…

encounter – встретиться, столкнуться

exact – точный, полный

to extend – расширять(ся)

to hold – held – held – удерживать, сохранять силу

identity – тождественное равенство

notation – обозначение

variable - переменная

Make your own sentences on the topic using the words above.

2. Fill in the gaps in the sentences with these words.

1) These are the problems you may … if you delete the PATH ….

2) An … is an equality relation A = B, such that A and B contain some … and give the same result when the … are substituted by any values (usually numbers).

3) A variable … constant is a variable in which you keep the same during the entire experiment.

4) An … … is considered to represent any one function of a set of functions.

5) In mathematics, an … differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering.

6) It will be a suitable … for showing interactions between different objects.

Read Text 2 and find out what the general solution means.

Text 2.

Review.

We begin by recalling the definition of a first-order differential equation. Consider an equation F (x,y,z)=0 in which not all the variables need occur, but at least z does. The equation

(1)

obtained by substituting for z , is a first-order differential equation. By a solution of (1) is meant any differentiable function f for which the equation

is true for every x in the domain of f. If f is a solution, we write

.

The general problem, given a differential equation, is to find all its solutions. A more specialized problem is to find a particular solution y=f(x) which has a specified value b at some specified number a, i.e., a solution for which b=f(a).

The simplest first-order differential equations are those of the type where f is some given function. Every solution of this differential equation can be written

y=∫f(x)dx+c,

for some real number c . Hence if c is left as an arbitrary undetermined constant of integration, we call ∫f(x)dx+c the general solution.

We next considered differential equations of the form , in which f and g are given functions. Equations of this type are called separable, since, if we use the fact that the derivative is equal to the ratio of two differentials, we can “separate” the expression containing x from that containing y by writing the equivalent differential equation

g(y)dy=f(x)dx.

Integrating both sides, we get the equation

∫g(y)dy=∫f(x)dx+c,

which defines the general solution y implicitly as a function of x. Note that the differential equation discussed above is a separable equation in which g(y) = 1.

Of special interest among separable equations is the first-order linear differential

equation , in which k is a constant. This is the type of differential equation which describes the rate of decay of a radioactive substance and also the rate of growth of bacteria in a culture. It can be solved without difficulty as a separable differential equation. However, this equation occurs sufficiently often and has such an obvious general solution that most people recognize it at sight. The general solution is y=ce^-kx.

*Source: Crowell and Slesnick's Calculus with Analytic Geometry The Dartmouth CHANCE Project Version 3.0.3, 5 January 2008

Comprehension questions:

1) What is a first-order differential equation?

2) What is a general problem for differential equations?

3) What is a more specialized problem?

4) What equations are referred to as separable?

5) What does it mean to define the solution implicitly?

6) Why are some separable equations of special interest?