d dtdyW La LI srt2 de le d 65 d SolutionThis is just repeate

d dt:dy=\'\'W La LI srt2 de le\" d 65 d

Solution

This is just repeatedly applying chain rule and product rule in Differential Calculus

It is required to express derivatives wrt X ( t = log X, so dt/dX = 1/X)) in terms of derivatives wrt t.

dy/dX = dy/dt .dt/dx (chain rule)

          = 1/X dy/dt              .....................................(1)

So d²y/dX² = d/dX (dy/dX) (by definition)

                   = d/dX(1/X dy/dt)..............................from (1)

                   = -1/X² dy/dt + 1/X d/dX (dy/dt)     (using product rule for differentiation)

                  = -1/X² dy/dt + 1/X d/dt (dy/dt) dt/dx (again using chain rule)

                   =-1/X² dy/dt + 1/X d²y/dt² 1/X

                   = 1/X² [ y\'\'(t)-y\'(t)], as required.