June 22nd, 2017, 11:59 AM  #11  
Member Joined: Jun 2017 From: India Posts: 65 Thanks: 3 
Yes, I was confused about it for a while too. But I thought mathematician was right in some way, because I am the noob. Quote:
Last edited by skipjack; July 18th, 2017 at 02:24 PM.  
July 18th, 2017, 10:11 AM  #12 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,957 Thanks: 801  
July 18th, 2017, 12:18 PM  #13 
Senior Member Joined: Jun 2015 From: England Posts: 740 Thanks: 208 
A few notes on differential equations might help, including showing the difference between ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs use the Roman letter d to denote the derivatives PDEs use the stylized Greek letter $\displaystyle \partial $ to denote the derivatives. ODEs have one independent and one dependent variable. PDEs have one dependent and 2 or more independent variables. A solution of either sort of differential equation is an expression or relationship between the variables, free of derivatives. For both, the order of the equation is the order of highest derivative present. For both, the degree of the equation is the degree of the highest order derivative (see example) So $\displaystyle \frac{{dy}}{{dx}} = x$ is a first order first degree ODE but note that this has one independent variable (x) and one dependent variable (y) $\displaystyle \frac{{dy}}{{dx}} = {x^2}$ is also a first order first degree ODE $\displaystyle \frac{{\partial y}}{{\partial x}} = k\frac{{\partial y}}{{\partial t}}$ is a first order first degree PDE Note that there are now two independent variables (x and t) and y is the dependent variable. $\displaystyle \frac{{{d^2}y}}{{d{x^2}}} = A\cos x$ is a second order first degree ODE $\displaystyle \frac{{{\partial ^2}\Omega }}{{\partial {v^2}}} = {\rm{Constant}}$ is a second order first degree PDE However if you look carefully this is 'missing' one independent variable. This can only happen if there is a second equation in which the second independent variable derivative is zero $\displaystyle \frac{{{\partial ^2}\Omega }}{{\partial {u^2}}} = 0$ $\displaystyle {\left( {\frac{{dy}}{{dx}}} \right)^2} = x$ is a first order second degree ODE. The solution of the first equation (an ODE) $\displaystyle \frac{{dy}}{{dx}} = x$ is $\displaystyle {\rm{y = }}\frac{1}{2}{x^2} + C$ Where C is an arbitrary constant. And here is the most important difference from PDEs You have to integrate the equation as many times as the order of the equation to produce a solution. Each integration of an ODE introduces an arbitrary constant, so a first order has one constant, a second order two arbitrary constants and so on. But for a PDE, each integration introduces an arbitrary function rather than a constant. For both types of equation (additional) boundary conditions are required to determine the arbitrary constants or functions. Normally, one condition per constant or function is required. Last edited by skipjack; July 18th, 2017 at 02:28 PM. 
July 18th, 2017, 12:57 PM  #14  
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 84  Quote:
I highly recommend Bleecker's Basic partial differential equations to the OP; it actually discusses this in quite some detail. Last edited by skipjack; July 18th, 2017 at 02:31 PM.  
July 18th, 2017, 01:42 PM  #15  
Senior Member Joined: Jun 2015 From: England Posts: 740 Thanks: 208  Quote:
Heaven knows there are few enough physically important PDEs we can solve analytically anyway. For most, we have to resort to numerical methods. I don't know Bleecker, but thanks for the reference. Last edited by skipjack; July 18th, 2017 at 02:30 PM.  
August 17th, 2017, 03:27 AM  #16  
Member Joined: Jun 2017 From: India Posts: 65 Thanks: 3 
Sorry for the late response; I almost forgot about this thread. Thanks a lot for explaining a lot of basic things studiot. I have few more doubts. Someone said many physics equations involve Ordinary and Partial differential equations. The only two physics equations worth learning to me looks like Maxwell equations and Schrödinger equation. Why I am a bit curious about these is because I see it in every physics documentary I see. And I would like to know more about those two equations. I was trying to make a few rough notes, and I don't know whether these things I wrote down and right. Quote:
Quote:
Last edited by skipjack; August 25th, 2017 at 08:37 AM.  
August 17th, 2017, 05:56 AM  #17 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 876 Thanks: 60 Math Focus: सामान्य गणित  
August 17th, 2017, 07:01 AM  #18 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 119 Thanks: 38 Math Focus: Algebraic Number Theory, Arithmetic Geometry  
August 17th, 2017, 07:44 AM  #19 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 876 Thanks: 60 Math Focus: सामान्य गणित  
August 25th, 2017, 08:05 AM  #20  
Member Joined: Jun 2017 From: India Posts: 65 Thanks: 3 
I have a few more questions to ask. I have been writing down a few notes and I am not sure whether this is right. Quote:
Quote:
In those equations, the variables could be about some measurements of electron, electricity, magnetism etc. Are all those things I wrote down right? Last edited by skipjack; August 25th, 2017 at 08:20 AM.  

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