Hi,(adsbygoogle = window.adsbygoogle || []).push({});

I'm working through 'Partial Differential Equations, an introduction' by Colton and am not finding it as clear as I hoped to.

I'm working through an example on how to solve a linear 1st order PDE.

I'll post Colton's example and Italic my questions:

Find the GS of

[tex]xu_x-yu_y+u=x[/tex]

The characteristic equation is

[tex]\frac{dy}{dx}=-\frac{y}{x}[/tex]

Integrating gives:

[tex]logy=-logx+c[/tex]

or

[tex]xy=\gamma[/tex]

Hence setting:

[tex]\zeta=x , \\

\eta=xy[/tex]

in our first order PDE yields:

[tex]\frac{\partial w}{\partial \zeta}+\frac{1}{\zeta}w=1[/tex]

Ok, my first question, why set[tex]\zeta=x , \eta=xy[/tex]

I don't really see how this relates to the original equation, and also eta doesn't seem to 'do' anything, why not set it to y, or xy^2 etx...i don't follow the logic

ctd...whose solution is:

[tex]w(\zeta,\eta)=\frac{\zeta}{2}+\frac{1}{\zeta}d(\eta)[/tex]

next question, this only works for[tex]d(\eta)=0[/tex]so what is the purpose of introducing it?

If anyone can help I'd be very grateful.

Rich

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# Homework Help: Help with PDE please

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