112 PRACTICAL MATHEMATICS 



Thus if y = e sin ~ la: 



Iog 6 y '-= sin" 1 x 

 Idy 1 



ydx 



and 



= z 



log e z = sin- 1 x - - log e (1 - x 2 ) 



+ 



__\ 



l-tf 2 / 



fill] f]1J 



We can therefore use the relation (1 x 2 ) -^ x -~ y = 0, 



' dx 2 dx 



combined with the condition that when x = 0, y = 1, to express y 

 as a series of terms of ascending powers of x. 



EXAMPLES VII 



y^2-vi ft /*i 



(1) If x = 3e-* sin 3t, prove that -77^ + 2 ^- + I0x = 0. 



{/ 6 CM/ 



(MA (\nJT 



(2) If x = 10 e- 2 *, prove that -rr- + 4 ^- + 4o? = 0. 



or o 



(l T* fi'V 



(3) If a; = 4 (e~ 2< - e~ 4< ), prove that -j-r- + 6 -j- + 8x = 0. 



ot or 



(4) If a? = 5 (1 + 40 e~ 2e , prove that -^ + 4 ^- + 4a; = 0. 



(5) If a? = 4 (3e- 2< - 2e- 4< ), prove that -^ + 6 -^ + 8a; = ( 



(If GJL 



d*y 



(6) If ?/ = (^ *, prove that -r - cos a; + y sin a? = 0. 



rttC Cti^ 



(7) li y= e^ sin &r, prove that ^ - 2a -j- + (a 2 + b 2 )y = 0. 



(8) If y = (?* cos bx, prove that - 2a + (a 2 + &% = 0. 



