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#### rannasquaer

##### New member

- Sep 5, 2020

- 6

In the differential equation:

\[ \frac{d^4y(x)}{dx^4}=\frac{1}{\text{EI}}q(x) \]

In which

\[ q(x)= P \delta(x-\frac{L}{2}) \]

P represents an infinitely concentrated charge distribution

The problem can be solved through developments in Fourier sine series, suppose that

\[ y(x)=\sum_{n=1}^{\infty} b_n \sin (\frac{n \pi x}{\text{L}}) \]

Demonstrate and explain step by step to obtain the equation below

\[ \delta(x-\frac{\text{L}}{2})= \frac{2}{\text{L}} \sum_{n=1}^{\infty} \sin (\frac{n \pi}{2}) \sin (\frac{n \pi x}{\text{L}}) \]