Differential equation question

[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 .

 

[stapel]

I think I understood the first part under the title of : meaning of differential equation...

What do you mean by "the first part"? "The first paragraph", up to the asterisk? Or something else?

...but I can't figure out how did he proceed

Please reply specifying (with a quote, perhaps, or a paragraph number) the point at which you become confused. Thank you!