x+siny=xy rArr siny=xy-x=x(y-1) rArr x=siny/(y-1)x+siny=xy⇒siny=xy−x=x(y−1)⇒x=sinyy−1
rArr d/dy(x)=d/dy(siny/(y-1))⇒ddy(x)=ddy(sinyy−1)
rArr dx/dy={(y-1)d/dy(siny)-sinyd/dy(y-1)}/(y-1)^2⇒dxdy=(y−1)ddy(siny)−sinyddy(y−1)(y−1)2
={(y-1)cosy-siny}/(y-1)^2=(y−1)cosy−siny(y−1)2
Since, siny=x(y-1)siny=x(y−1), we have,
dx/dy={(y-1)cosy-x(y-1)}/(y-1)^2=(cosy-x)/(y-1)dxdy=(y−1)cosy−x(y−1)(y−1)2=cosy−xy−1
:. dy/dx=(y-1)/(cosy-x).......................................................(star)
rArr y'(cosy-x)=y-1
:. d/dx{y'(cosy-x)}=d/dx(y-1)
:. y'd/dx(cosy-x)+(cosy-x)d/dx(y')=dy/dx-0=y'.
:. y'{d/dxcos y-d/dx(x)}+(cosy-x)y''=y'.
:. y'{(-siny)y'-1}+(cosy-x)y''=y'.
:. (cosy-x)y''=y'(y'siny+1)+y'=y'(y'siny+2).
=(y-1)/(cosy-x){((y-1)siny)/(cosy-x)+2}.....................[by (star)]
:. y''=(y-1)/(cosy-x)^2{((y-1)siny)/(cosy-x)+2}
rArr (d^2y)/dx^2=y''=[(y-1){(y-1)siny+2(cosy-x)}]/(cosy-x)^3.
Enjoy maths.!
