« НазадПродовжити »
This appears to be the solution of the equation, but it does not satisfy it unless C, = 0, when it becomes
which is only a particular integral, and therefore incomplete.
This arises from our implying in the use of equation (4) that nbn + qbn-1 = 0 is generally true, whereas the equation
(n − 1) (nbn + qon-1) = 0, derived from the auxiliary equation
dax + 9 dx = 0, shews that b, is not necessarily connected with bu, since it may be satisfied by n = 1.
To complete the solution, we have from (2) which is always true
do = - bos and from (4) which is true for n= - 1, we have
These quantities are independent of an, 22, &c., therefore writing C, for a, as it is an arbitrary constant, .
is a particular integral of the proposed equation, and the complete solution is
y=C(1-) + C(1+) -
PARTIAL DIFFERENTIAL EQUATIONS.
Sect. 1. Linear Equations with Constant Coefficients.
By the method of the separation of symbols the integration of Linear Partial Differential Equations is reduced to the same processes as those for the integration of ordinary differential equations of the same class. Hence the theory which is given in the beginning of Chap. iv. is equally applicable to the present subject, and it is unnecessary to repeat it here; I shall therefore content myself with referring to what has been previously said in the Chapter alluded to, adding that every differential equation of this class between two variables has an exact analogue among partial differential equations of the same class, and that the form of the solution of the latter is the same as that of the former. On this point one remark may be made which is of considerable importance in the interpretation of our results. As in the solution of ordinary differential equations we continually meet with expressions of the form
Cear, so in partial differential equations we shall find expressions of the form
in which the arbitrary function takes the place of the arbitrary constant. Now as the preceding formula is the symbolical expression for Taylor's Theorem, we know that
Hence, in the solution of partial differential equations, arbitrary functions of binomials play the same parts as arbitrary constants multiplied by exponentials do in equations between two variables.
Now supposing v to be the independent variable, and .. a constant, with respect to it, by the Theorem given in Ex. (11), Chap. xv. of the Diff. Calc. this is equivalent to
or, effecting the integration, and adding an arbitrary function of y, instead of an arbitrary constant,
or, as the form of ø is arbitrary, we may for
(-x) write (ay – bæ), so that
It is obvious that if we had taken y for our independent variable, and considered as a constant with respect to it, we should have had
as a constant