CHAP. VI.] FORM OF THE GENERAL SOLUTION. 303 



We can write, instead of V one of the roots V 2 , 3 , &c., and 

 compose by means of them a more general value expressed by 

 the equation 



Z-kf9i r / x \ 



= a l e~ x* I cos f 2 -^ Ju l sin qjdq 



g%# 3 r 

 A-a / 



cos f 2^7^ sin c + &c. 



!, a 2 , a 3 , &c. are arbitrary coefficients : the variable q dis 

 appears after the integrations, which should be taken from q = 

 to q = TT. 



315. To prove that this value of v satisfies all the conditions 



f -, &quot; WfJWM . IT- .^Sf^SJ*^ ****M&amp;gt;*iB- 



oi the problem and contains the general solution, it remains only 

 to determine -the coefficients a lf 2 , a z , &c. from the initial state. 

 Take the equation 



v = af m ^u^ + a 2 e~ mit u 2 + a/r m ^ u 3 + &c., 



in which w 1? w 2 , w 3 , &c. are the different values assumed by the 

 function u, or 



- m x z m* x* 

 ~ + ~ 



77? 



when, instead of -y-, the values ^, ^ 2 , ^ 3 , &c. are successively sub- 



K 



stituted. Making in it t = 0, w T e have the equation 

 V =* a^fj -f a 2 u 2 + 3 w 3 + &c., 



in which F is a given function of x. Let &amp;lt; (x) be this function ; 

 if we represent the function u i whose index is i by &amp;gt;/r (xtjff^ we 

 have 



^ (x) = a^ (a? V^) + a.^ (x Jg} + a 3 ^ (a; v/^ 3 ) + &c. 



To determine the first coefficient, multiply each member of 

 the equation by c^ dx, cr^ being a function of x, and integrate from 

 x = to x = X. We then determine the function cr^ , so that after 

 the integrations the second member may reduce to the first tenn 

 only, and the coefficient a l may be .found, all the other integrals 



