Correct way to solve this differential equation

[ehilge]

Homework Statement

Suppose a beam of length L is supported at x=0 and x=L. if the load per unit length is given by:

w(x)=(w₀x)/L

Then the differential equation for the deflection, y(x), of the beam is given by

a*y⁴(x)=(wx)/L

Where a and w₀ are constants

a)find the Fourier series for the odd periodic extension
b) Find a particular solution for this differential equation

Homework Equations

The Attempt at a Solution

So, I know how to do part (a), what worries me about this problem is (b). It seems like to solve the equation, I should just be able to divide the a over and integrate 3 times to recover y(x). To me this makes perfect sense, however, the question is in a section mainly over Fourier transforms so it seems unlikely that the question wouldn't include fouriers. Also, these questions are normally constructed so you have to use part (a) to find (b). So, is there any reason why I can't solve (b) in the simple manner I described and if so, how could I go about using Fourier transforms to solve the problem?
Thanks!

 

[mighty2000]

Thank you for your question. It is true that you can use integration to find a particular solution for this differential equation. However, using Fourier transforms can also be a useful approach for solving this problem. Here's how you can use Fourier transforms to solve part (b):

1. Take the Fourier transform of the given differential equation. This will convert the differential equation into an algebraic equation in terms of the Fourier transform of y(x).

2. Solve the algebraic equation for the Fourier transform of y(x). This will give you the Fourier transform of the particular solution.

3. Take the inverse Fourier transform of the particular solution to get the actual solution for y(x).

Using Fourier transforms can be a more efficient and elegant way to solve this problem. I hope this helps. Good luck with your studies!