Question

If y = 3 cos (log x) + 4 sin (log x), prove that: x²y₂+xy₁+ y =0.

Answer 1

 Note: y₂ represents second order derivative i.e.\(\frac{d^2y}{dx^2}\) 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 \(\frac{d^2y}{dx^2}\)

As, \(\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})\)

So, lets first find dy/dx

Let, log x = t

∴ y = 3cos t + 4sin t …………….equation 2

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