Separation of variables on 2nd order ode

[koab1mjr]

Hi all

Quick one, if one had an equation y' = x on could simply separate the variables and integrate. Now it the equation y'' = x you would use separation of variables what drives this?

Also

y'' =0. Is the same as. y''dx =0 dx
Why is this legal?Thanks in advance

 

[HallsofIvy]

Well, you can always multiply both sides of an equation by the same thing! So, yes, y''= 0 is the same as y''dx= 0dx= 0. And you can now integrate both sides of the equation, with respect to x, to get y'(x)= C where C is a constant of integration. But you did not really need to do that. You know, I hope, that if the derivative of a function is 0, then the function is a constant: if the second derivative of y is 0, the first derivative is a constant.

 

[koab1mjr]

Ok re read what I wrote and my main question is not clear

Why is

D^2y/Dx^2 = 0 not the same as d^2y =0 dx^2

On one side u get a line on the other u get something else what is the rigorous explanation