Solve PDE Method of Characteristics help

[lackrange]

The problem: Solve for u(x,y,z) such that

xu_x+2yu_y+u_z=3u\; \;\;\;\;u(x,y,0)=g(x,y)

So I write

\frac{du}{ds}=3u \implies \frac{dx}{ds}=x,\; \frac{dy}{ds}=2y\;\frac{dz}{ds}=1 .

Thus u=u_0e^{3s},\;\;x=x_0e^{s}\;\;y=y_0e^{2s}\;\;z=s+z_0
but from here I can't figure out what to do, there are several ways I can write s...I have only done the method of characteristics before with two variables, and those are pretty much the only examples I can find. Can someone help please?

 

[HallsofIvy]

The problem: Solve for u(x,y,z) such that

xu_x+2yu_y+u_z=3u\; \;\;\;\;u(x,y,0)=g(x,y)

So I write

\frac{du}{ds}=3u \implies \frac{dx}{ds}=x,\; \frac{dy}{ds}=2y\;\frac{dz}{ds}=1 .

Thus u=u_0e^{3s},\;\;x=x_0e^{s}\;\;y=y_0e^{2s}\;\;z=s+z_0
but from here I can't figure out what to do, there are several ways I can write s...I have only done the method of characteristics before with two variables, and those are pretty much the only examples I can find. Can someone help please?

If you have done the method of characteristiced with two variables it pretty much generalizes to three variables directly- write out the characteristics in terms of x, y, and z and use those to define new variables.

If x= x_0e^s and y= y_0e^{2s}, you have s= ln(x/x_0)= (1/2)ln(y/y_0) so that ln(x/x_0)- ln(y^{1/2}/y_0^{1/2})= ln((y_0^{1/2}/x_0)(x/y^{1/2}= 0 which then gives x/y^{1/2}= constant. Each curve with different constant is a characteristic. With z= s+ z_0 that gives also s= z-z_0= ln(x/x_0), have x/x_0= e^{z-z_0}= e^{-z_0}e^z so x= (constant)e^z or xe^{-z}= constant. Putting s= z-z_0= ln(y^{1/2}/y_0^{1/2}), we get y^{1/2}e^{-z}= constant.

We want to use those "characteristics" or "characteristic curves" as axes: let u= x/y^{1/2}, v= xe^{-z}, and w= y^{1/2}e^{z}.