This looks simple enough, but we find that we cannot express the RHS in the form of 'x-factors' and 'y-factors', so we cannot solve by the method of separating the variables.

In this case we make the substitution y = vx, where v is a function of x. So y = vx. Differentiate with respect to x (using the product rule): dy/dx = v+x(dv/dx)
then you substitute for y and dy/dx in the main equation thereby transforming the Equation in to separable type.

Note: dy/dx =(x+3y)/2x is an example of a #homogeneous #differential #equation. This is determined by the fact that the total degree in x and y for each of the terms involved is the same (in this case, of degree 1). The key to solving every homogeneous equation is to substitute y = vx where v is a function of x. This converts the equation into a form which we can solve by separating the variables.

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