190 EXAMPLES OF THE CHANGE 



9. Suppose u to be a function of r and 



then if 



d*u d*u d*u d z u _ 



+ + ...... + ~ ' 



shew that 



d*u n ldu_ 

 dr* r dr 



10, Given x = acos<f>, ;?/ = > sin <, express 



'1 + f W 

 \dx) ) 

 , 2 in terms of <f>. 



Result 



. 

 ab 



d'-v ^-e^dy 4w 2 y 



11. Transform jj + i-jr^ Z+.jfT&p-* mto an 



equation in which t shall be the independent variable, 

 having given x = log >v/(tan t). 



Besult -fi + tfy = Q. 



12. Change the independent variable from y to x in 



d 3 u d*u 2 du A 



^-3 - 4 tan y ^- a + 2 tan 2 y = 0, supposing tan y = x. 



Sesult (i + , T g + 2a; ( 1 + a: . ) g + 2^ = 0. 



13. Transform ~+ ( ? } into an expression in which y is 



dx \dxj 



the independent variable. 



72 



14. Given x = t + t 2 , transform -^ into an expression in 



ctt 



which x is the independent variable. 



