Page images
PDF
EPUB

The integral of this equation is deduced from that in

d Ex. (13) of Chap. V., by putting a

This gives

dy 2

2 y y'

f (y ax).

for q.

[ocr errors]

(23) The equation

d"%

[ocr errors]

pm dix

dxn-1

dye

0, dum

dy may be integrated by the same method as that in Ex. (14) of

d? Chap. vi., by changing k into – a and putting arbitrary functions of y instead of the arbitrary constants. Thus if m = 2, we have

X = F (y + ax) + f (y - ax), so that the integral of

[ocr errors][ocr errors]

2p dx
Xdx

[ocr errors]
[ocr errors]
[ocr errors]

P

1

% = w2p+1

[ocr errors]
[ocr errors]

= 0

da*

dy
i d
is

{F (y + ax) + f (y – ax)}.

dx
Hence if p = 2, the integral of

4 de

a? dv adx

dy is x=3{F(y+aw)+f(y-ax)}- 3av {F'(y+aw), f'(y-ax)} + a’x{F"(y + ax) +f" (y – ax)}.

(dx do
(24) Let
+

% = 0. dx dy

dy)

(x + y)2 Assume y + x = U, y - x = 0, when the equation becomes

2 dx 2 du? dv2

u The integral of this by Ex. (18) is

{$'(u + v) +%'(u – v)} - {p (u + v) +y (u – v)}.

[ocr errors]
[blocks in formation]
[ocr errors][ocr errors][merged small][merged small][merged small][merged small]

Hence,

1

[ocr errors]

dv

1 {p' (y) + V (2x)} {Q (24) ++ (2x)}.

(x + y)
Non-linear equations of the form

dy
P
+Q

0 ().R,

dy P, Q, R being functions of x and y, may be transformed into linear equations by assuming

d:

= d.x'.
Φ (8)
dz

ху
(25) Let

+ y dc

dz

dy

[ocr errors]
[merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small]

We might with advantage have applied the same transformation to the equations in examples (1), (3), and (4), as it is generally convenient to reduce the factor of x to two terms.

Sect. 3. Equations involving the differential coefficients

of . in powers and products.

dx

dx

dy

dp

dy

d x

If the equation be of the first order make

P, = 9,

dx and from the given equation find q in terms of p, x, y, z, and substitute this value in the equation dp dq

d q +9 р

0,

(1)

d & d x which will then become an equation of the first order between four variables. The value of p found by integrating this, with the corresponding value of q will render dx = pdx + qdy,

(2) a complete differential, and this being integrated will give the value of s. The integral of the first equation will involve an arbitrary constant (a); and the integral of the second will introduce another (6), which is to be considered as an arbitrary function of (a); and we shall thus obtain an integral of the form

f (x, y, x, a) = q (a), from which a is to be eliminated when a specific meaning is assigned to 0.

Lagrange, Mémoires de Berlin, 1772, p. 353. (1) Let p + q* = 1, or q = (1 - po), р dp da

P

dp
do
(1 – p)! da

(1 – po)! dz Substituting these values in equation (1) it becomes dp dp

dp + (1 - p) dx This equation is integrable if we can integrate the system of equations d p=0, pdx - dx = 0, (1 – po)?dz - dy = 0.

da

[ocr errors]

dx

+ P

da

= 0.

dy

so that

The first gives p=l, whence q = (1 – a”), and

= adx + (1 - a')!dy,

z = ax + (1 - a') y + P(a). If we differentiate this with respect to a we obtain the equation

(1

( = X -
(1 – ao)

y + $'(a), between which and the preceding we may eliminate a when o is specified. (9) Let

Pa = 1.
The equation in p to be integrated is
dp 1 dp

2 dp
dy

p2 dx
whence dp = () and p = a. The final iniegral is

Y
P = ( x + + P(a).

+

+

0;

pdz

a

[merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

= ax + aʼy + (a). (6) Let

q = x P + p. The integral is

x = x 6%++ 62 (4+4) + (a).

'do ed &
(7) Let
wo zy

= c".

d x
This may be put under the form

[ocr errors]

= c".

[ocr errors]

By assuming

d' ,
'

' +; the equation becomes

'do

dy The integral of this found by the same method as in Ex. (2) is

[ocr errors][merged small]

mtnty

[ocr errors]

m

n

1

[ocr errors]
[ocr errors]

+

m-a

m +n

n-B a" xm

y m+n+y

+ + (a).

n-ß an
When m = a, a' = log x, when n = B, y = log y,

when m +n + y = 0, z' = log x.

« PreviousContinue »