MHB What is integration of y/(x^2-y^2) dx

[r-soy]

Hi all

can please explaine to me what is integration of y/(x^2-y^2) dx

step by step ...

 

[alyafey22]

Re: what is integration of y/(x^2-y^2) dx

Hey Ahmed :

I think you have already studied partial fraction decomposition when you learned about integration .

Here we can treat y as constant hence, we don't need to worry about it , because we are integrating w.r.t x ...

we know by the difference of two squares that :

$$x^2-y^2=(x-y)(x+y)$$

$$\frac{y}{(x-y)(x+y)}=\frac{A}{x-y}+\frac{B}{x+y}$$

Hence we have the following :

$$y = A(x+y) + B(x-y) $$

Now we need to find both A and B so do the following :

1- put x= y so the equation becomes :

$$y = 2y\, A$$ , $$ A =\frac{1}{2}$$

2-To find B we put x=-y

$$y = -2y \,B $$ , $$ B =\frac{-1}{2}$$

$$\frac{y}{(x-y)(x+y)}=\frac{\frac{1}{2}}{x-y}+\frac{\frac{-1}{2}}{x+y}$$

Now can you integrate the right hand side ?

 

[r-soy]

Re: what is integration of y/(x^2-y^2) dx

1/2ln(x-y) + -1/2(x+y)

 

[MarkFL]

Re: what is integration of y/(x^2-y^2) dx

Hi all

can please explaine to me what is integration of y/(x^2-y^2) dx

step by step ...

Did you check out the Heaviside cover-up method I pointed you to the other day for partial fraction decomposition?

 

[alyafey22]

Re: what is integration of y/(x^2-y^2) dx

1/2ln(x-y) + -1/2(x+y)

No, this isn't correct , you are missing an absolute value and a ln in the second part !

Also , don't forget that there should be a constant which is a function of y ...

 

[r-soy]

Re: what is integration of y/(x^2-y^2) dx

1/2ln(x-y) + -1/2ln(x+y) + c

- - - Updated - - -

Did you check out the Heaviside cover-up method I pointed you to the other day for partial fraction decomposition?

Yes

 

[alyafey22]

Re: what is integration of y/(x^2-y^2) dx

1/2ln(x-y) + -1/2ln(x+y) + c

Still , you are missing the absolute value , also C here is a function of y it is is usually wirtten $$\phi(y)$$

 

[Prove It]

Re: what is integration of y/(x^2-y^2) dx

Hi all

can please explaine to me what is integration of y/(x^2-y^2) dx

step by step ...

Has this question come from trying to solve a differential equation, which would make y a function of x? Or are you doing a "partial integral", in other words, holding y constant while trying to integrate with respect to x?

 

[I like Serena]

Re: what is integration of y/(x^2-y^2) dx

Hi all

can please explaine to me what is integration of y/(x^2-y^2) dx

step by step ...

Another approach.
If you have a list of derivatives of trigonometric hyperbolic functions, you should have:
$$\frac{d}{dx} \text{ artanh } x = \frac 1 {1-x^2}$$

 

[Prove It]

Re: what is integration of y/(x^2-y^2) dx

Another approach.
If you have a list of derivatives of trigonometric functions, you should have:
$$\frac{d}{dx} \text{ artanh } x = \frac 1 {1-x^2}$$

Hyperbolic functions...

 

[soroban]

Hello, rsoy!

$\displaystyle\int \frac{y}{x^2-y^2}\,dx$

Since $y$ is treated as a constant $b$, we have: .$\displaystyle b\int\frac{dx}{x^2-b^2}$There is a standard integration formula: .$\displaystyle \int \frac{du}{u^2-a^2} \:=\:\frac{1}{2a}\ln\left|\frac{u-a}{u+a}\right|+C $Therefore: .$\displaystyle b\left(\frac{1}{2b}\right)\ln\left|\frac{x-b}{x+b}\right|+C \;=\;\frac{1}{2}\ln\left|\frac{x-y}{x+y}\right|+C$

 

[Prove It]

Hello, rsoy!


Since $y$ is treated as a constant $b$

The OP has not specified if this is actually the case...

 

[MarkFL]

The OP has not specified if this is actually the case...

I believe your earlier suspicion that this arose in the process of solving an ODE was correct, i.e., that $y$ is actually dependent upon $x$.