Temperature distribution - PDE's

[Anabelle37]

Homework Statement

Consider a square of unit length which is insulated on the sides x=0, x=1 and y=0 and has the temperature distribution u(x,1)=x along the side y=1. find the temperature distribution u in the square.

Homework Equations

The Attempt at a Solution

So I have the boundary conditions:
du(0,y)/dx = 0; du(1,y)/dx = 0; du(x,0)/dx = 0; u(x,1)=x (where these are partial derivatives)

I am unsure where to go from here...do i solve laplaces equation? if so I don't know how to put the boundary conditions in terms of the separation of variables method.

please help!

 

[HallsofIvy]

Yes, the equilibrium distribution of temperature on a plate satisfies $\nabla^2 u(x, y)= 0$.

If you write u(x, y)= X(x)Y(y), then
$$\frac{\partial u}{\partial x}= Y(y)\frac{dX}{dx}$$
so
$$\frac{\partial u(0, y)}{\partial x}$$$$= Y(y)\frac{dX(0)}{dx}= 0$$
so either Y(y)= 0 for all y, which gives only the trivial solution, u= 0 for all x and y or
$$\frac{dX(0)}{dx}= 0$$
That's your boundary condition on X.

Similarly for the other two.

 

[Anabelle37]

ok thanks,
do i partial du/dx for the other 2? or du/dy?