I9II-] LINEAR DIFFERENTIAL EQUATIONS. 281 



Solving these partial differential equations in regular order for 

 Fq, fj, Vo, v., ■■■, substituting these values in the assumed expres- 

 sion for V, and finally making 6^ unity, the result 



F= <^(/>' + ^ <f>'(t) '^' + ^, <}>'V) f^+'-- 

 (B) 



is a solution of Fourier's differential equation for all values of 

 <^(^) and 6(t) for which V either contains a finite number of terms 

 or is an infinite series uniformly convergent both for x and for t. 

 Solutions of the differential equation when (j>(t) =o correspond- 

 ing to several values of ^(0 are as follows — 



II </>(/) = O, 

 (i) d{i)=A, V=A, 



(2) d{f) = Af, ^=^{' + ^)' 



(3) ^(/) = At\ F= A (/^ + J.' + ^, ) , 



2 \{2Kt) \\{2Kif 



3-y^ 1 



'^ t\{2Ktf "y 

 (5) e(t) = ArK V=-^^, 



3 



2 ! 2A7 ' 4 ! {2KtY 



3 '^' ] 



^,)eit) = At-K V=At-^[.--^-^^ + 



6 ! {2Kty 

 3-5 ■«^* 



4! (2A'/)2 



3-5-7 __^' 

 6! {2Ktf "^ ■ 



] 



