Question on How energy equation is presented in Diff form

[thunderspeed]

Hi guys ,
I'm currently reading a book and I faced this with DEs:
In the 5 attached images you'll find the textbook paragraph I'm asking about .
I think I understood the first part under the title of : meaning of differential equation but I can't figure out how did he proceed with the same concept on applying it to the energy equation .

Please help . Thanks in advance .

 

[Simon Bridge]

Welcome to PF;
That's a lot of reading you want people to do on the off-chance they will understand your problem.
Which of the many paragraphs are you having trouble with?
What is it about the paragraph that is giving you trouble?
i.e. please let us know how you are thinking so we can best respond to your problem.
Alternatively, please attempt to express your problem as a specific question.