Question

Let y=e^(cos^-1x) for modx<1.
Show that (1-x^2)(d2y/dx2)=x(dy/dx) + y.
By further differentiation, show that e(cos^-1x) ≈ e^a(1 - x + 0.5x^2 - bx^3) where a and b are real constants to be determined.

I managed to show the first part but I don't get how to do the second part. How do you "further differentiate" the equation from the first part?

Answer 1

I think they want you to approximate it with the Taylor series, which involves y", y"', etc.