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eshen
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.
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
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.