The dependent variable changes with respect to the independent one, thus dependent = function of independent.

To solve the equation is to find a function of the independent variable, which has this special property that its second derivative + first derivative to the power of 3 - 3 times the independent variable + 2 times the function itself = 0.

To solve an ODE means to find the function as a clear expression or if it is hard to find one, you can use numerical methods to construct it. This is not easy and from what I see from your questions you should do more learning.

Not always the case. You cannot always separate x and y from the given differential equation. The ones where you can separate x and y terms, are of the form
F(x)G(Y)dx+f(x)g(y)dy=0.

Ofcourse, it is also possible using a transformation or integrating factor. It is important to also know the forms of different differential equations etc. I would get Ross:Differential Equations.

that is why you read a differential equations book. Heck, even read Paul's Differential Notes if you do not have access for a book or find a free one on the internet. It is not hard to google. What you took seems to be a baby numerical analysis course..

I have to question your understanding on a few factors. How did you take this course without knowing what what derivatives are, what ordinary equations are, and how to solve them. Yet, your syllabus has such methods as Eulers, Picard, Runge Katta, and Interpolation?

Did you not read the requirements for the course? Or was is a class where the instructor assumed no prior knowledge and simplified his/her lectures?

If you want to understand what these terms, then read an ODE book. You will have understood what these things are in atleast 1 month of studying...

mine was a three year Bsc , 6 semester computer science course ...

in the second semester , after some introduction to basic electronics ... we had this subject called ... "computer oriented numerical methods in c programming language "

the texts were only referring to the mathematics and the programming side of it was missing from most of the usual textbooks we had ...

the main programming language text ... The C Programming Language

was missing numerical methods in it ...

so most people didn't had much clue either ... where this syllabus was heading to ...

most still doesn't have much clue ...

its only sometime later , for learning purposes ... i started to read these books instead ...

Yes all belong to a differential equations for students who have no exposure to it. Some books have a partial differential sections, but it is better to get a separate book on that subject.

I think OP is missing the basic concepts and needs to catch up with them. When I stumble upon such problem, as already suggested, I use google to find an appropriate way to handle it.