Partial Diff qn

fredrick08

1. The problem statement, all variables and given/known data
Does the following differential equation for u(x,y) have solutions which have the form of a product of functions of each independent variable:

[tex]\partial[/tex]²u/[tex]\partial[/tex]x[tex]\partial[/tex]y=u


[b]3. The attempt at a solution[/]

=>[tex]\int[/tex][tex]\int[/tex]u dx dy =uxy=f(x)f(y) ???

fredrick08

im not sure if that even makes sense, but if u(x,y)=f(x)f(y) and d*f(x)/dx=u/constant and d*f(y)/dy=constant... i think, i don't know im confused..

Dick

Set u(x,y)=f(x)*g(y). What is your partial derivative in terms of f(x), f'(x), g(y), and g'(y)?

fredrick08

so [tex]\partial[/tex]²u/[tex]\partial[/tex]x[tex]\partial[/tex]y=f(x)g(y)??
so in terms of f'(x) and g'(y) im not sure..

Cyosis

You have an equation that equates the derivative of u to u itself. You can already guess what u will be, but lets not guess.

Setting u(x,y)=f(x)g(y), then the equation becomes [tex]\frac{\partial^2 f(x)g(y)}{\partial x \partial y}=f(x)g(y)[/tex]. Now what Dick asks you to do is to calculate [tex]\frac{\partial^2 f(x)g(y)}{\partial x \partial y}[/tex].

Hint:

[tex]
\frac{d}{dx}f(x)= ?
[/tex]

In terms of f'(x)

Dick

I actually meant u(x,y)=f(x)g(y). There's no reason why the function of x would have to be the same as the function of y. And yes, evaluate the derivative.

Cyosis

Sorry, that was sloppy of me I actually meant f(x)g(x). I will fix it.

fredrick08

hmm d/dx of f(x).. isn't that just f'(x)??.... is it something like the integral of f'(x)g'(y)??

fredrick08

so... [tex]
\frac{\partial^2 f(x)g(y)}{\partial x \partial y}
[/tex] =[tex]\int[/tex][tex]\int[/tex]f'(x)g'(y) dx dy??

Cyosis

What you've written down now is the equivalent to [itex]\frac{d}{dx}f(x)=\int f'(x) dx[/itex]. Which is wrong. Lets test it, f(x)=x then [itex]\frac{d}{dx}x=1 \neq \int 1 dx=x+c[/itex].

This is right, so if you evaluate [tex]\frac{\partial^2 f(x)g(y)}{\partial x \partial y}[/tex] you get....?

fredrick08

u get f'(x)g'(y)?? Im confused becasue when u take partial of f(x), u get d/dy(d/dx*f(x))=0??

Cyosis

Yes that's correct, but you're not taking the partial of f(x), you're talking the partial derivative of f(x)g(y).

[tex]
\frac{\partial^2 f(x)g(y)} {\partial x \partial y}=\frac{\partial}{\partial x} f(x) \frac{\partial}{\partial y}g(y)=f'(x)g'(y)
[/tex]

fredrick08

ok so since f(x)g(y)=f'(x)g(y)=u(x,y)=??

Cyosis

You forgot a prime it should be [itex]f'(x)g'(y)=f(x)g(y)=u(x,y)[/itex]. You can instantly see which function satisfies the equation, but you can also separate the variables.

fredrick08

e^x+y?? so f(x)=e^x and g(y)=e^y same as the derivativeS?

Cyosis

That would be a solution indeed, you're on the right track. it is not the general solution however. The general solution will involve integration constants and a constant in the argument of the exponential. Can you see how to integrate the differential equation?

fredrick08

e^xy?? so f(x)=e^x and g(y)=e^y same as the derrivativeS?

srry this was an accident