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How do I solve the differential equation of the type $$ x u_x + y u_y = f(x, y) $$ For example, let $f(x, y) = xy$. Using following method, I found $F(x,y) = F(x/y)$ is solution to homogeneous equation

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What method can I use to find the particular solution of the given DE? When $f(x,y) = xy$, I put $$ u_y = cx \implies u(x, y) = cxy $$ Putting this on DE, I found $c=1/2$. Is there general way to solve for particular solution of PDE?

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You can change to new variables $s=x/y$ and $t=y$ (for example).

(Let's say that we're looking for solutions in the region $y>0$, to avoid having to worry about division by zero.)

Then $u_x=u_s s_x + u_t t_x = u_s/y$ and $u_y=u_s s_y + u_t t_y = -u_s \, x/y^2 + u_t$, so your PDE becomes $$ x \, u_s/y + y \, (-u_s \, x/y^2 + u_t) = f(x,y) , $$ i.e., $$ u_t = \frac{f(x,y)}{y} . $$ If you now express the right-hand side in terms of the new variables, you get something of the form $$ u_t = g(s,t) , $$ and now you can simply integrate with respect to $t$, to get $$ u(s,t) = G(s,t) + F(s) , $$ i.e., $$ u(x,t) = G(x/y,y) + F(x/y) , $$ where $G$ is the particular solution you asked about, and $F$ is the homogeneous solution that you already knew.

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