I am quite embarrassed to ask this question, as I know i have lost track of the concept here, but Ill nevertheless ask it. I was going through Mathematical methods for physicists (pg 333), and there was an example:(adsbygoogle = window.adsbygoogle || []).push({});

"Solve $$y'+(1+\frac{y}{x}) = 0$$"

My problem is,

(a) when you put the equation into the general form of an exact equation, and get:

$$\psi = \int^{x}_{x_0} (x+y)dx + \int^{y}_{y_0} x_0dy$$,

$$\psi = \frac{x^2}{2} + xy + C$$,

Why do you treat y and x as constant wrt to each other when integrating?

More specifically, why is $$\int^{y}_{y_0} x_0dy = x_0 \int^{y}_{y_0} dy$$, and the same thing with y in the first integral? There is no explicit assumption of dependence of x on y and vice versa, So I'm guessing it is because P and Q of the exact solution are functions of 2 variables, but you integrate only wrt one.

(b) Why are the rest of the terms($$-x_0^2/2 , x_0y_0)$$, of the integral constant?

I would like to stress that it is NOT my objective to SOLVE the equation, but to get my concept of integrating different variables wrt to each other clear. In brief, I want to know how to integrate expressions such as (x+y) and xy wrt to x(or y). And I am not talking about double integrals here. If you don't have y=f(x) as an expansion, i.e. x and y are mere variables when integrated wrt each other, how to integrate the expression is my query. Also, from the book it seems that they have taken $$\int(x+y) dx = \int x dx + \int y dx$$,i.e "distributed the integrand". How can you do that, since the integral is only once, how can you make 2 integrals out of it? In short, does integration have some sort of distributive rule and each term of the expression in brackets integrated separately wrt the variable in question(dx in this case)? If so, then WHY? or am I missing something? Please elaborate.

Thanks in advance!!!

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# Integrating ##d\psi=(x+y)dx +x_0dy##

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