Question

1. Consider the differential equation x4y′′ − x3y′ = 8.
(b) Show that Ax^2 + 1 + B is a solution to the equation, where A and B are x^2
constants.

Answer 1

I would make the coefficient in front of the y'' a 1, and then use u=y' to reduce it to a first order differential equation. And then use a simple integrating factor technique

Answer 2

What is part (a)?

Answer 3

One way I can think of is to make the left side looks like the Cauchy-Euler equation..
\[ x^4y′′ − x^3y′ = 8\]\[ x^2y′′ − xy′ = \frac{8}{x^2}\]Set the left =0
\[\lambda (\lambda -1) - \lambda = 0\]\[\lambda = 0, 2\]\[y_c = c_1x^2+c_2\]And then find the particular solution.

Answer 4

Thanks a bunch!