Mathematics. ''On integral equations connected tnith differentia/ 

 equations. By J, Droste. (Communicated by Prof. J. C. Kluyver). 



(Communicated in the meeting of March 29, 1919). 



^1. As is shown by Hilbert in his second paper on integral 

 equations (Gött. Nachr. 1904, p. 2J3 sq.) there exists a connection 

 between linear integral equations with symmetrical kernel and linear 

 differential equations of the second order with linear homogeneous 



conditions between r/. and — at the ends of the interval. Taking e.g. 



dx 



in the interval (0,J) the equation 



■é + >"' = '- ■■■■■■■■ m 



(I being a constant, it may asked to determine the function ^(.t!) so 

 as to satisfy the differential equation and a condition at both ends 

 of (0,1). This is only possible for certain values of f/, the socalled 

 characteristic numbers. Having chosen a value ,a,, for which the 

 problem fails to have a solution, it will on the contrary be possible 

 to find a symmetrical function K{x,y), which, as a function of /■, 

 satisfies the differential equation and the conditions at the ends of 



the interval and which moreover has its derivative — — -^ — disconti- 



nuous for d' = y. The characteristic numbers and functions of the 

 kernel K{x,y) (being the numbers n and the functions ^/^(.r). for which 

 the equation 



1 



'fix) =z (I j K{.v,y) <f(t/)dy . . ..... (2) 







is valid) are the same values of n and the same functions (f{.r) that 

 solve the problem of the differential equation. 



It appears, that the kernels considered above, viz. those satisfying 

 the differential equation, are not the only ones to have </(.??) for their 

 characteristic functions. For the purpose of the reduction of the 

 problem of the differential equation to another problem it is unne- 

 cessary to consider other kernels. But when an integral equation is 

 given, it may be useful to have a method that enables us to know 

 whether the kernel has or has not for its characteristic functions solutions 



