While working on a problem in game theory, I'm stuck at a problem which requires me to solve the following linear first order PDE on $K$ independent variable:
$\sum_{k=1}^K(\frac{\partial u}{\partial y_k}y_k \sum_{j=1}^n \lambda_{jk}a_j) = r(a_i,\bar{y})$
I've tried looking at the method of characteristics but I don't really know the boundary conditions. I found this method to find the general solution https://www.math.ualberta.ca/~xinweiyu/436.A1.12f/PDE_Meth_Characteristics.pdf but am not really sure how to extend it to n -dimensions.
I have the following relation:
$\frac{dy_k}{y_k \sum_{j=1}^N\lambda_{jk}a_j} =\frac{du}{r(a_i,\bar{y})} $ for all $k=1:K$
Here, $r(a_i,\bar{y}) = c + \sum_{k=1}^Ky_kc_k$
Any help or direction would be great.