If y = 3 cos (log x) + 4 sin (log x), prove that: x2y2+xy1+ y =0.
Note: y2 represents second order derivative i.e. and y1 = dy/dx
Given,
y = 3 cos (log x) + 4 sin (log x) ……equation 1
to prove: x2y2+xy1+ y =0
We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative.
Let’s find
As,
So, lets first find dy/dx
Let, log x = t
∴ y = 3cos t + 4sin t …………….equation 2
………….equation 3
……...equation 4
Again differentiating w.r.t x:
Using product rule of differentiation we have
Using equation 2,3 and 4 we can substitute above equation as:
Multiplying x2 both sides:
∴ x2y2+xy1+ y =0 ………..proved