
Differential Equations Ordinary and Partial Differential Equations Math Forum 
 LinkBack  Thread Tools  Display Modes 
February 3rd, 2007, 06:38 PM  #1 
Newbie Joined: Nov 2006 From: Cubs Town Posts: 5 Thanks: 0  Partial Differential Equation
Verify that this function is everywhere harmonic. (e)^x Cos(y) 
February 3rd, 2007, 08:44 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,833 Thanks: 2161 
We need to differentiate the function twice with respect to x (treating y as though it's a constant), then start again and differentiate twice with respect to y (treating x as though it's a constant). In the process, we note that our results are valid everywhere (the functions used are differentiable everywhere). The resulting second derivatives are easily found and are (e^x) cos(y) and (e^x) cos(y). Since they are defined for all values of x and y and their sum is zero, the original function is everywhere harmonic. That was easy, wasn't it? 

Tags 
differential, equation, partial 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
How to integrate this partial differential equation  JulieK  Differential Equations  0  December 28th, 2012 07:12 AM 
Partial Differential Equation  mathbalarka  Differential Equations  0  September 19th, 2012 09:14 PM 
THE PARTIAL DIFFERENTIAL EQUATION  mathbalarka  Differential Equations  3  March 30th, 2012 05:21 PM 
partial differential equation  sixflags11804  Differential Equations  1  September 29th, 2008 01:26 PM 
Partial Differential Equation  WananoMath  Differential Equations  7  February 1st, 2007 10:42 AM 