Question

If `x^2 y=a cos⁡`x, where `a` is a constant, show that `x^2 (d^2 y)/(dx^2 )+4x dy/d +(x^2+2)y=0`?

Answer 1

We have:

`x^2 y=a cos⁡x`

If we differentiate implicitly wrt `x` and apply the product rule then:

`(x^2)(d/dx y) + (d/dx x^2)(y) = -asinx`

`:. x^2dy/dx + 2xy = -asinx`

Differentiating again we wgt:

`(x^2)(d/x dy/dx) + (d/dx x^2)(dy/dx) + (2x)(d/dxy) + (d/dx2x)(y) = -acosx`

`:. x^2 (d^2y)/(dx^2) + 2xdy/dx + 2xdy/dx + 2y = -acosx`

`:. x^2 (d^2y)/(dx^2) + 4xdy/dx + 2y + acosx = 0`

And using the initial equation:

`x^2 (d^2y)/(dx^2) + 4xdy/dx + 2y + x^2 y = 0`

Leading to:

`x^2 (d^2y)/(dx^2) + 4xdy/dx + (x^2+2) y = 0 \ \ \ \` QED