Is it possible to do separation of variables in this manner? dx = ?(? + 1)(? − ?)
This the equation I have and I am attempting to use separation of variables in order to integrate it
dx = ?(? + 1)(? − ?)
So, the integral of 1/(? + 1)(? − ?) dx = the integral of k dt
Then,
A/?+1 + B/?−? will therefore = 1/ (?+1) (?−?)
and (after multiplying the LHS and RHS by (x+1)(n-x), I achieved
?(?−?) +?(?+1) =1 (Eq. 1)
I have been told that the following is a method used I should use to solve for A and B
(?−?)?+?+??=1
⇒?=? and ?+??=1
⇒?+??=1⇒?=1/(1+n)⇒?= 1/(1+n)
However, I am confused about how it is known that B=A.
Instead, I thought it was possible to substitute 'useful' values for x in order to find a and then b. For example, sub in x=-1 to find A, and substitute x=n to find B.
So my question is, how does A=B and also, it is possible to let x=n in order to find B (I.e. use 'useful' values to cancel out B when finding A and A when finding B?)
Thank you.
This the equation I have and I am attempting to use separation of variables in order to integrate it
dx = ?(? + 1)(? − ?)
So, the integral of 1/(? + 1)(? − ?) dx = the integral of k dt
Then,
A/?+1 + B/?−? will therefore = 1/ (?+1) (?−?)
and (after multiplying the LHS and RHS by (x+1)(n-x), I achieved
?(?−?) +?(?+1) =1 (Eq. 1)
I have been told that the following is a method used I should use to solve for A and B
(?−?)?+?+??=1
⇒?=? and ?+??=1
⇒?+??=1⇒?=1/(1+n)⇒?= 1/(1+n)
However, I am confused about how it is known that B=A.
Instead, I thought it was possible to substitute 'useful' values for x in order to find a and then b. For example, sub in x=-1 to find A, and substitute x=n to find B.
So my question is, how does A=B and also, it is possible to let x=n in order to find B (I.e. use 'useful' values to cancel out B when finding A and A when finding B?)
Thank you.
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