Solve the Liner governing equation of bridge structure stage forming equation.

1. I need the procedure to get the result of the linear governing equation.
2. Here are provided with the attached file, the method, and the result. I need to understand how they calculate and get results.
3. As you can see the picture 5 they use to solve the problem.
4. Pictures 1, 2, 3, and 4 are the problem and data.
5. I have attached the Pdf file, where gives the whole description and problem and result. I just need to understand the procedure of calculation and get results.