The first transformation gives

(ax+by' — aa−bß + c) dx' + (a'x' + b'y' — a'a−b'ß+c') dy' =0,

whence if a and ẞ be determined by the conditions

aa+bB = c,

a'a + b'ß = c',

we shall have the homogeneous equation

(ax' + by') dx' + (a'x' + b'y') dy' = 0.

Making then y = vx' we find

adx + bdy = dx', a'dx+b'dy=dy',

whence determining dx and dy, the proposed equation assumes the homogeneous form

(b'x' — d'y') dx' — (bx' — ay') dy' = 0.

Both these transformations fail if ab' — a'b=0. But in this case, since b' = =

in the form

a'b

the proposed equation may be expressed

α

(ax+by+ ac") dy = 0,

α

(ax+by+c) dx + a(

(

and the variables will be separated if we assume ax + by = 2, and then adopt either z and x or z and y as the new variables.

These transformations are linear, and by one of the two the proposed equation is usually solved.

[For another method see the Supplementary Volume, Chapter XIX, Arts. 1 and 2.]

10. The linear differential equation of the first order and

degree

dy + Py = Q

dx

(21),

Pand being functions of x, admits of being solved. When Q=0 the solution is obtained by separating the variables; and when is not equal to 0, a solution may be founded upon that of the previous and simpler case.

It must be observed that the linear equation (21), when reduced to the form

(Py− Q) dx + dy = 0,

falls under the general type, Mdx + Ndy = 0.

1st, When Q=0, we have

Dividing by y, in order to separate the variables,

Therefore, log y=- Pdx+c, which gives

C being an arbitrary constant substituted for e. It has been already observed that a function of an arbitrary constant is itself an arbitrary constant; see Art. 4.

2ndly, To solve the linear equation (21) when Q is not equal to 0, let us assign to the solution the general form (22) above obtained, but suppose C to be no longer a constant but a new variable quantity—an unknown function of x, which must be

determined in accordance with the new conditions to which the solution must be subject.

Substituting then the above expression for y in (21), and observing that, since C is now variable, we have

c being an arbitrary constant. Substituting this generalized value of C'in (22), we have finally

It will be observed that if Q=0, the above solution is reduced to the form (22) before obtained.

The method of generalizing a solution above exemplified is called the method of the variation of parameters, the term parameter, by an extension of its use in the conic sections, being applied to denote the arbitrary constants of the solution of a differential equation. It is only, however, in certain cases that this method is successful. It is always legitimate to endeavour to adapt a solution to wider conditions by a transformation, which, like the above, only introduces a new variable instead of an old one, or a new and adequate system of variables in the room of a former system. But it is not always that the equations thus obtained are, as in the above example, easier of solution than those of which they take the place.

Therefore y = (+1) ((+1) + 0}.

2

+o}.

= €* (x + 1)”.

[Pdx = − n log (x+1),

: ƒ3å3 = (x + 1)",

fers Qdx =[edx =

e.

y = (x+1)" (e2 + c).

P and Q being functions of x, are reducible to a linear form. For, dividing by y", we have

The solution of this equation, which is identical in form with that of Ex. 1, is

General solution by development.

12. In the earlier portion of this Chapter it was established, by considerations founded upon the nature and interpretation of the equation

Mdx+Ndy = 0,

that it implied the existence of a primitive equation between x, y, and an arbitrary constant. The examples of finite solution which have been given above, illustrate this truth. But