Laplace's Equation: d^2 T/dx^2 + d^2 T/dy^2 = 0

Bob_2013

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I have been posed the following question;

The steady-state temperature distribution T on a flat sheet in thexy-plane obeys Laplace’s equation:

. . . . .\(\displaystyle \dfrac{\partial^2 T}{\partial x^2}\, +\, \dfrac{\partial^2 T}{\partial y^2}\, =\, 0\)

(a) Demonstrate that each of the functions
(i) T1(x, y) = e−2ycos 2xand
(ii) T2(x, y) = ln px2 + y2 is a solution to Laplace’s equation.
(b) For each of these two temperature functions, determine the rateof change of the temperature if moving from the location (0, −1)towards the origin.

and I don't know how to preform part a) so I was wondering if someone might be able to walk me through the process?

cheers :D
 

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I have been posed the following question;

The steady-state temperature distribution T on a flat sheet in thexy-plane obeys Laplace’s equation:

. . . . .\(\displaystyle \dfrac{\partial^2 T}{\partial x^2}\, +\, \dfrac{\partial^2 T}{\partial y^2}\, =\, 0\)

(a) Demonstrate that each of the functions
(i) T1(x, y) = e−2ycos 2xand
(ii) T2(x, y) = ln px2 + y2 is a solution to Laplace’s equation.
(b) For each of these two temperature functions, determine the rateof change of the temperature if moving from the location (0, −1)towards the origin.

and I don't know how to preform part a) so I was wondering if someone might be able to walk me through the process?

cheers :D
First calculate the partial derivatives of the proposed solutions (functions) and then use those in the given differential Equation and see if the equation is satisfied.
 
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I have been posed the following question;

The steady-state temperature distribution T on a flat sheet in thexy-plane obeys Laplace’s equation:

. . . . .\(\displaystyle \dfrac{\partial^2 T}{\partial x^2}\, +\, \dfrac{\partial^2 T}{\partial y^2}\, =\, 0\)

(a) Demonstrate that each of the functions
(i) T1(x, y) = e−2ycos 2xand
(ii) T2(x, y) = ln px2 + y2 is a solution to Laplace’s equation.
(b) For each of these two temperature functions, determine the rateof change of the temperature if moving from the location (0, −1)towards the origin.

and I don't know how to preform part a) so I was wondering if someone might be able to walk me through the process?

cheers :D
Please use parentheses! I can guess that "e-2y" is supposed to be e^(-2y) but there is no way of telling if, by "ln px2+ y2" you mean ln(px^2)+ y^2 or ln(px^2+ y^2) or ln(p) x^2+ y^2 or ln(p))(x^2+ y^2).

To illustrate what Subhotosh Khan means, if T1(x, y)= e^(-2y) cos(2x) then T1_x= -2 e^(-2y)sin(2x) and T1_(xx)= -4 e^(2x) cos(2x). Then T1_y= -2 e^(-2y) sin(-2x) and T2_(yy)= 4e^(-2y)sin(-2x).

What do you get when you put those second derivatives into Laplace's equation?
 
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