Question

If y = xsinx, then prove that, x²(d²y/dx²)  – 2x(dy/dx) + (x² + 2)y = 0.

Answer 1

Given y = xsinx

⇒ dy/dx = xcosx + sinx

⇒ d²y/dx² = cos x – x sinx + cosx

⇒ d²y/dx² = 2cosx – xsinx

Now, x²(d²y/dx²)  – 2x(dy/dx) + (x² + 2)y

x²(2cosx – xsinx) – 2x(x cosx + sinx) + (x² + 2)x sinx

= x²(2cosx – xsinx + x sinx – 2cosx) + x(2sinx – 2sinx)

= x²(0) + x(0) = 0