90 DIFFERENTIAL AND INTEGRAL CALCULUS. 



Ex. (a 2 # 2 ) -, z . x -j- + wfy 0, where x = a cos 0, 



<&/ _ cly dx _ 1 c?y 

 ~~ 



dO ' 



d 2 y d / 1 <fy\ ^0 



therefore -j^ = ^ r = ,>, -7- . 



c?ic 2 dO \ a sm ^ dd] dx ' 



f? 2 ?/ _ 1 d' 2 ?/ 



" = "*" >2 ~ 



/ 2 >2\ 2 va 



therefore (a 2 ~x) -f^= a sm 2 ^ T - 2 = . - . /, 

 ; ^x 2 da? 



6?y /i l ^ 



-x--=-acos6 



-i 

 a sin 6 dO 



C I 2 !!\ ^^ ^V ^^ 



therefore (a - *) ^ -* - = ^ , 



and the transformed equation is -~ + 7/i' 2 ?/ = 0. 



(x\ 

 -} ; 



rfy dy dd - 1 ^/y 



therefore -f- = -^ -r- = ., 2 . A -^ ; 



dx dd dx v(or or) c?^ 



therefore Vt^ - **} ^ = - ^ ; 



therefore 



^ 2 V( 2 - ! )^'~ dff'dx~ ^(a'-x')dd^ 



*'-*'> S--*l=8; 



so that in this case the second method gives the result more 

 readily, but there is not usually much to choose between 

 them. 



As an example of changing both dependent and inde- 

 pendent variables, take the equation 



