Need help with partial differential equation

[Needhelp2013]

Homework Statement

Given that z = √3x/y show that:

Homework Equations

∂²z/∂x∂y = ∂²z/∂y∂x

The Attempt at a Solution

 

[bossman27]

Both sides of that equation are the same, did you mistype something?

 

[Needhelp2013]

Yes I did, that's it sorted now.

 

[bossman27]

Ahh, as I suspected. Have you made an attempt yet?

 

[Needhelp2013]

Thanks for spotting that.No I am pretty lost to be honest. I am fairly new to this. I believe I am working with the Laplace equation though

 

[bossman27]

Thanks for spotting that.No I am pretty lost to be honest. I am fairly new to this. I believe I am working with the Laplace equation though

I think you're thinking too hard, you know how to take a partial derivative I assume?

 

[bossman27]

The equation under "Relevant Equations" is what you're trying to show, correct?

 

[Needhelp2013]

Yes the equation under "Relevant Equations" is what I'm trying to show. There could be a simple solution but I'm missing it. I tried substituting what 'z=' into the laplace equation but that didnt get me anywhere.

 

[bossman27]

I tried substituting what 'z=' into the laplace equation but that didnt get me anywhere.

Exactly what I meant by thinking too hard. All that it's asking you to do is show that if you take the derivative of z with respect to x and then y, it's the same as taking the derivative of z with respect to y and then x. In other words, show:

$\frac{\partial}{\partial y}(\frac{\partial z}{\partial x}) = \frac{\partial}{\partial x}(\frac{\partial z}{\partial y})$

Does that make any more sense now?

 

[Needhelp2013]

I am just trying to work out how to read the latex way of writing equations. Ill get it. Thanks for that.

 

[Needhelp2013]

Exactly what I meant by thinking too hard. All that it's asking you to do is show that if you take the derivative of z with respect to x and then y, it's the same as taking the derivative of z with respect to y and then x. In other words, show:

$\frac{\partial}{\partial y}(\frac{\partial z}{\partial x}) = \frac{\partial}{\partial x}(\frac{\partial z}{\partial y})$

Does that make any more sense now?

Is your equation meaning this ∂y/zx = ∂x/zy

any other help would be great because I am struggling to realize where to go next.

 

[bossman27]

You can't read that equation? It should read: ∂/∂y(∂z/∂x) = ∂/∂x(∂z/y)

 

[Needhelp2013]

You can't read that equation? It should read: ∂/∂y(∂z/∂x) = ∂/∂x(∂z/y)

Thanks again.Is the next step now to bring the ∂z over to the other side and that is it complete?

 

[bossman27]

No, figure out the parts in the parentheses first. That is, find ∂z/∂x and ∂z/∂y.

You don't seem to be familiar with taking partial derivatives, so here's a quick explanation:

∂z/∂x = the derivative of z with respect to x; treat y as if it were a constant (i.e. just a number). Simply take the x derivative like you would for dz/dx, if y was just some number.

∂z/∂y is exactly the same, just switch x and y in my instructions.