Integration using separation of variables

Integrate the following integral using separation of variables :

v_{x} = (1/3)v -(8/9) (v_{x} is the partial derivative of v w.r.t x) and v is a function of two variables.

Can someone please show me the steps involved for separating the variables for this question ?

Please Help.

Re: Integration using separation of variables

How many variables is v a function of?

Re: Integration using separation of variables

v is a function of two variables , x and y

Re: Integration using separation of variables

Since y does not appear in the equation, treat as an equation in x only.

$\displaystyle \frac{dv}{dx}= \frac{1}{3}(v- \frac{8}{3})$

$\displaystyle \frac{dv}{v- \frac{8}{3}}= \frac{dx}{3}$

Integrating

$\displaystyle ln\left|v-\frac{8}{3}\right|= \frac{x}{3}+ f(y)$

where the "constant of integration" can be any function of y only.

We can solve that for $\displaystyle v(x, y)= \frac{8}{3}+ e^{x/3}F(y)$

where, since f(y) can be an arbitrary function of y, $\displaystyle F(y)= e^{f(y)}$ is also an arbitrary function of y.

Re: Integration using separation of variables

But in my book the answer is v(x,y) = 8/3 +(1/3)e^x/3 F(y).

Where do you think that extra factor of 1/3 came from ? or is it incorrect ?