Higher mathematics for students of chemistry and physics by Mellor J

Author:Mellor, J[oseph] W[illiam] [from old catalog]
Language: eng
Format: epub
Tags: Mathematics, Chemistry, Physical and theoretical
Publisher: London, New York [etc.] Longmans, Green and co.
Published: 1902-03-25T05:00:00+00:00

state A to the state B, In fact the total work done in the passage from A \Ki B and back again, is represented by the area APBQ (page 183). In order to know the work done during the passage from the state A to the state B, it is not only necessary to know the initial and final states of the substance as defined by the coordinates of the points A and B, but we must know the nature of the path from the one state to the other.

Similarly, the quantity of heat supplied to the body in passing from one state to the other, not only depends on the initial and final states of the substance but also on the nature of the transformation.

All this is implied when it is said that ^'dW and dQ are not perfect differentials ". Although we can write

'dx^y ~ 'by'dx* we must put, in the case of W or Q,

' or —

^x'by ^ 'dy'dx ' 'dxby 'dy'dx ' Therefore the partial diiferentiation of x with respect to ^, furnishes a complete differential equation only when we multiply through with the integrating factor /x, so that

where x and y may represent any pair of the variables y^ v, $,

The integrating factor is proved in thermodynamics to be equivalent to the so-called GamoVs function (see Preston's Theory of Heat), To indicate that dW and dQ are not perfect differentials, some writers superscribe a comma to the top right-hand corner of the differential sign. The above equation would then be written,

d'Q = dU -\- d'W,

§ 122. Linear Differential Equations of the First Order.

A linear differential eqtuition of the first order involves only the first power of the dependent variable y and of its first differential coefficients. The general type is,

% + Py = ^ (1)

where P and Q may be functions of x, or constants.

§ 122. HOW TO SOLVE DIFFERENTIAL EQUATIONS. 297

We have just proved that ^"^ is an integrating factor of (1),

therefore

e^'^{dy + Pydx) « eJ''*'Qdx,

is an exact differential equation. The general solution is,

yef^^ = ^e^'^Qdx + C. . . (2)

The linear equation is one of the most important in applied mathematics. In particular cases the integrating factor may assume a very simple form.

In the following examples, remember that e^^' ^ x, .'. if \Pdx = logo;, e^**^ = x.

Examples. —(1) Solve (1 + x'*)dy = (m + xy)dx. Reduce to the form (1)

And we obtain

dy X _ m

/C xdx ^^ = - jrr^ = - 4iog(i + x») = - log v(i + ^)'

Remembering log 1 = 0, loge s 1, the integrating factor is evidently,

log^/'-' = log 1 - log N^m?, or d/«« = 'J^TT^' Multiply the original equation with this integrating factor, and solve the resulting exact equation as § 119, (4), or, better still, by (2) above. The solution: y = mx -^ C ,J{1 + a;*) follows at once.

(2) Ohm^s law for a variable current flowing in a circuit with a coefficient of self-induction L (henries), a resistance R

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