On solving PDE using separating the variable.

[A Dhingra]

hi..

with refrenence to http://www.math.uah.edu/howell/MAPH/Archives/Old_Notes/PDEs/PDE1.pdf
page 7,
“Observe” that the only way we can have

formula of t only= formula of x only​

is for both sides to be equal to a single constant.

here I do understand that for these to being equal requires them to be constant, because changing either t or x would affect either side of the formula breaking the inequality, so to be equal they are supposed to be a constant. (please correct me if i go wrong anywhere..)
Now what i am not sure about is that these two formulas can also be equal to each other for a specific value of x and t, is this anything to do with the constant that they will be equal to??

It is said that the constant say k is arbitrary, suggesting these two sides are equal for more than one value of x and y... how is this said?

(I gave the link so that i don't have to explain what i am asking, because i don't know the correct terms and that might turn confusing.. no copyright violation intended)

 

[HallsofIvy]

hi..

with refrenence to http://www.math.uah.edu/howell/MAPH/Archives/Old_Notes/PDEs/PDE1.pdf
page 7,
“Observe” that the only way we can have

formula of t only= formula of x only​

is for both sides to be equal to a single constant.

here I do understand that for these to being equal requires them to be constant, because changing either t or x would affect either side of the formula breaking the inequality, so to be equal they are supposed to be a constant. (please correct me if i go wrong anywhere..)
Now what i am not sure about is that these two formulas can also be equal to each other for a specific value of x and t, is this anything to do with the constant that they will be equal to??

I'm not sure what you are asking here. If X(x)= T(t) for all x and t, then, yes, they must be equal to the same constant for all x and t, which, of course, includes any "specific value of x and t".

It is said that the constant say k is arbitrary, suggesting these two sides are equal for more than one value of x and y... how is this said?

(You mean "t" not "y", right?) What do you mean "for more than one value of x and t"?? You just said they were equal for all x and t. That surely includes more than one!

(I gave the link so that i don't have to explain what i am asking, because i don't know the correct terms and that might turn confusing.. no copyright violation intended)

 

[A Dhingra]

I'm not sure what you are asking here. If X(x)= T(t) for all x and t, then, yes, they must be equal to the same constant for all x and t, which, of course, includes any "specific value of x and t".

Okay i got it..
If X(x) = T(t) for all x and t, and there is no relation or interdependence between x and t other than this, then X(x)= constant just as T(t). And this constant is constant for all values of x and t... Thanks for the quick help.