1. Given x⁵y" + 2x⁴ y' + (1-x³) y = 0 (1-1)

where y' and y" are respectively dy/dx and d^2y/dx^2.

a) Does this equation have regular or irregular singular points?

b) Respectively, use 1/z, dy/dz and d^2y/dz^2 to replace x, dy/dx and d^2y/dx^2 so that equation (1-1) becomes f(z, dy/dz ,d^2y/dz^2) = 0; explain that when x is sufficiently large, letting x = 1/z can be a good approach.

c) Find the solution of your new equation f(z, dy/dz ,d^2y/dz^2) = 0 and

d) With the help of your solution z =1/x, find the workable interval of x.