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VI. 



A NEW METHOD OF SOLVING LEGENDEE'S AND BESSEL'S 

 EQUATIONS, AND OTHEES OF A SIMILAE TYPE. 



By JOSEPH EOOEESON COTTEE, M.A. 



Read May 27. Ordered for publication June 26. Published December 27, 1907. 



The method of solution explained in this paper is intended for the purpose 

 of obtaining the complementary function in the case of any ordinary differ- 

 ential equation in which all the terms but one can be integrated by the 

 process of rendering the equation an exact differential equation by multi- 

 plication by a suitable factor. With regard to the outstanding term, the 

 successive multiplications and integrations are represented symbolically, and 

 a symbolical operator finally arrived at which gives the solution in the form 

 of a series. The method possesses the advantage that it gives all the particular 

 integrals of the complementary function at once; and it gives the solution also 

 of those cases of Legendre's and Bessel's equations in which the general solution 

 fails. 



Among the commonest types of equations which can be integrated by the 

 process indicated are linear equations with constant coefficients, and homo- 

 geneous equations. The former are integrated by multiplying by some power 

 of e^, and the latter by multiplying by some power of o^. For instance, using 

 tlie symbol D to represent djclx, the equation 



D'li - -IBy + 2/ = 



is readily integrated by multiplying by e'^. This gives 



e^B\j - 2e-''Dy + r^?/ = 0, 



which, being integrated, gives 



e~^Dy - e'^y = A ; 



and this, without further multiplication, gives 



e'-'y = A + Bx. 



Equations of this type are usually solved by a slightly different mode of 

 procedure ; thus, to solve f{D) y = 0, the algebraical equation f{z) = is first 

 solved, and then the solution of the differential equation can be written down. 



