Solving n-dimensional first order linear pde

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.