I don't want to type this out, so I took a picture of it.
I don't understand the beginning differentiating part. [Why f(x) x f'(y)]
How does letting y=0 give you f'(y)=A (I know A is just a variable, using that as a value of something)
How does f'(x)=Af(x)?
I don't get the substituting x=y=0 part on the bottom, why x=y=0?
As for theassuming their logic and notation is correct (I have my suspicions...) notice that you are trying to findWhy substitute y = 0?
.
But we are not given , it's .
But if , then
, which is what we're trying to find.
Remember that , so
.
For any function of y, this function depends ONLY on y. Substituting ANY value for y ALWAYS gives you a constant.
is one such function of y.
Since 0 is a constant, is also a constant.
Thus , where A is a constant.
Therefore
.
Is that a bit clearer?
Here's another way maybe a little clearer. Assume that clearly it's true if . Take the natural log's of both sides
.
Now differentiate wrt to x and then y. So
.
Then set either . You now have an ODE which can be integrated easily. Pretty much like the later part of the book.
Sorry for bothering you guys with these noob questions.
I have another one, and this is probably a very dumb question.
Where did the 1 come from in the beginning when they differentiate both sides?
nvm. Damn, that's a noob question. I thought they're differentiating . They're actually differentiating