Question

Hey... I was thinking back to partial differential equations... And I was wondering... when using separation of variable...why is it that when we separate the variables both expressions equal a constant.... For example... pretend we have \[y u_x-x u_y=0\]...And we get to .... \[\text{ Let } u=X(x) \cdot Y(y)\] And then we get to this part... \[\frac{X'}{Xx}=\frac{Y'}{Yy}\] \[\text{ We are able to say this is equal to some constant.}\]

Answer 1

That last line is what I can't remember why that is so.

Answer 2

for fun I will finish...
\[\frac{X'}{Xx}=\frac{Y'}{Yy}=k \text{(why?)} \\ \text{ anyways } \\ \frac{X'}{Xx}=k \text{ and } \frac{Y'}{Yy}=k \\ \frac{dX}{X}=xk dx \text{ and } \frac{dY}{Y}=y k dy \\ \ln|X|=\frac{kx^2}{2}+C_1 \text{ and } \ln|Y|=\frac{ky^2}{2}+C_2 \\ X=c_1 e^{\frac{kx^2}{2}}=\text{ and } Y=c_2 e^{\frac{ky^2}{2}} \\ \text{ and so } u(x,y)=c_1 c_2 e^{\frac{k}{2}(x^2+y^2)}\]

Answer 3

maybe it is because whatever they equal can't be a function of both x and y
since X'/(Xx) is just a function of x
and Y'/(Yy) is just a function of y
and a constant function is the only function which you say is a function of any variable