i have few doubts about differential equations and numerical methods ...
in a differential equation question ... you are given an instantaneous rate of change...
and you are supposed to find the function that gives , this instantaneous rate of change

i am trying to understand some things in plain english ...
in a differential equation question,you are given an instantaneous rate of change of an object
The gradient of the curve dy/dx is the instantaneous change of y with respect to x ,the rate at which y is changing as x is changing
and you are supposed to find the function that gives , this instantaneous rate of change

wasn't i supposed to find the function that gives ,this instantaneous rate of change ??
The answer is the function that has that rate of change
The answer is a function that has the instantaneous rate of change that you were given ...

now i have four functions and four instantaneous rate of change ??
what does this mean ??
does it mean that i have four functions that has the instantaneous rate of change that i was given

You have two equations with a different character: one (##{dy\over dx}=x^2-3##) is of the type ## y'= f(x) ## and the other is of the type ##y'= f(y)## (I use the abbreviation ##y'## for the derivative ##{dy\over dx}## ).

## y'= f(x) ## is solved if you can find a primitive of f : a function ##F(x)## for which ##F'= f##.

There is the small matter of the integration constant: If you know your rate of change but you don't know where you start, then you can not determine where you will be next.

In other words: With every differential eqiuation you need something to deal with the integration constant (one for every "quote") . For example a function value at x = 0 (or a function value at any other point).

Your first equation isn't accompanied by such a condition, so you can't get further than ## y = F(x) = x^3/3-3x + K##.

Your second differential equation does have a condition: ##y(0) = 1##. You solve numerically by approaching y(0) as a linear function for a small step:$$y(0+\Delta x) \approx y(0) + \Delta x {dy\over dx}$$ when all you really know is $$ \lim_{h\downarrow 0} { y(0+h) - y(0) \over h } = {dy\over dx} \ .$$
When you take a few steps you get a sequence of approximated function values, but not a function (in the sense of an exact prescription what to do with x to get y).

So for ##\ y'= y\ \ \& \ \ y(0) = 1 \ ## you have found four function values, not four functions.

thanks a lot .. there is so much good information in that post ... i have already learned a lot from your post ... now i think its just a matter of googling the proper words from your post ..

even differential equation is kind of getting interesting ...

few more basic questions ...

i dont understand this properly ...

most of the time , the original question is ...

find the function that gives ,this instantaneous rate of change

Nice picture; taken from inside the book ? (I recognize Rotterdam, though...)

Seems to me you still have trouble understanding what you did for ##y'=y## so let's step back and treat ##y'= x^2-3## the same way. Add initial condition y(0) = 0 so it yields a unique solution.

So the exact solution is a function ##y = x^3/3-3x## and the numerical approach is a table of aproximated function values.

(I took steps of 0.1 so the difference between exact and numerical becomes visible).

In fact, for your other equation ##y'=y## and ##y(0)=1## an exact solution can also be found. Use the same method (everything with y on one side gives ##{dy\over y}=dx##, then integrate and apply the initial condition). But you'll learn that one soon enough.

its from a book called "numerical methods for ordinary differential equations " ...i did a horizontal flip of that image with ms paint .. Mod note: Why? How does that add any information to your question? Also, you have posted an image of a bridge twice, which also adds nothing of substance to what you're asking.
few more re arrangements in a way that i can understand this ...

i am going to take a little bit of time to read and re read this ...

there is so much good information again on this one ...

let me see what i can learn from this , and i don't know where to move to from here ...

again , thanks a lot ...

if i have more doubts i will come back and post more ...