- #1
lackrange
- 20
- 0
The problem: Solve for u(x,y,z) such that
[tex] xu_x+2yu_y+u_z=3u\; \;\;\;\;u(x,y,0)=g(x,y) [/tex]
So I write
[itex]\frac{du}{ds}=3u \implies \frac{dx}{ds}=x,\; \frac{dy}{ds}=2y\;\frac{dz}{ds}=1 .[/itex]
Thus [tex]u=u_0e^{3s},\;\;x=x_0e^{s}\;\;y=y_0e^{2s}\;\;z=s+z_0 [/tex]
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?
[tex] xu_x+2yu_y+u_z=3u\; \;\;\;\;u(x,y,0)=g(x,y) [/tex]
So I write
[itex]\frac{du}{ds}=3u \implies \frac{dx}{ds}=x,\; \frac{dy}{ds}=2y\;\frac{dz}{ds}=1 .[/itex]
Thus [tex]u=u_0e^{3s},\;\;x=x_0e^{s}\;\;y=y_0e^{2s}\;\;z=s+z_0 [/tex]
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?