SECT. IV.] A DIFFERENTIAL EQUATION. 405 



be v = cos (tD*) &amp;lt;j&amp;gt; (#,#), denoting ^ by D; for we deduce from 

 this value 



d* 



572 -V = - -j-4 V. 



dt dy? 



The general value of v, which can contain only two arbitrary 

 functions of x and y, is therefore 



v = cos (ZD 2 ) (a?, y) + W, 



and TF = f dt cos (*Z&amp;gt; 2 ) -^ (#, y). 



Jo 



Denoting u by /(a?, y, 0, and ^ by / (a;, y, ), we have to 

 determine the two arbitrary functions, 



* & y) =/ (^ y* )&amp;gt; and ^ (^ y) =/ te y&amp;gt; o). 



403. If the proposed differential equation is 

 tfv d*v d*v 



_ 

 - 



we may denote by D$ the function -y + -gj so that 



or Z) 2 ^&amp;gt; can be formed by raising the binomial ( -j- a + -p 2 j to the 



second degree, and regarding the exponents as orders of differen- 



d?v 

 tiation. Equation (e) then becomes -^ + D z v = 0; and the value 



of v, arranged according to powers of t, is cos (tD) &amp;lt;f&amp;gt; (x, y) ; for 

 from this we derive 



7 . ^ /, or ^^ + -y- 4 + 2 , 2 , 2 + -7-4 = 0. 



ar cfo cfar dx dy dy 



The most general value of v being able to contain only two 

 arbitrary functions of x and ?/, which is an evident consequence of 

 the form of the equation, may be expressed thus : 



v = cos (tD) &amp;lt;/&amp;gt; (x, y) + 1 dt cos (tD} f (#, y). 



