Basic PDE Help: Simplifying the Confusing Concepts | MathBin

[jomomma]

http://mathbin.net/906

cant figure this one out

 

[tiny-tim]

Welcome to PF!

x\frac{\partial u}{\partial x}+2y\frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} = 3u

cant figure this one out

Hi jomomma! Welcome to PF!

(type "tex" instead of "EQ" on this forum )

What have you tried?

 

[HallsofIvy]

What do you mean by "figure this out"? Did you stare at it expecting the answer to suddenly pop into your head or did you DO something like writing down the equations of the characteristics?

If the latter, please show us what you did!

 

[jomomma]

I tried doing characteristics but i am getting confused with the u_z term. Heres what i do

<br /> <br /> \[<br /> \frac{dx}{dt}&amp;=&amp;x\Rightarrow x=x_0e^t<br /> \]<br /> \[<br /> \frac{dy}{dt}&amp;=&amp;2y\Rightarrow y=y_0e^2t<br /> \]<br /> \[<br /> \frac{dz}{dt}&amp;=&amp;1\Rightarrow z=z_0 + t<br /> \]<br /> \[<br /> \frac{du}{dt}&amp;=&amp;3u\Rightarrow u=u_0e^3t<br /> \]<br /> <br />

from there i try to put y and x into the equation for u

<br /> u=xy=x_0y_0e^3t<br />

what do i do with z and \varphi(x,y)

 

[jomomma]

could someone verify whether the correct answer is

<br /> <br /> u(x,y,z)=\varphi(xe^{-z},ye^{-2z})e^{3z}<br /> <br />

 

[NoMoreExams]

Can't you plug it in and check like you would with any DE?

 

[jomomma]

yes, i can and i have, but i wanted third party confirmation, because i may not have included all solutions or may have made a mistake