Non-homogeneous Differential Equation Solutions

Date: 02/26/2001 at 18:48:49
From: Elliott Zimmermann
Subject: Homogeneous and non-homogeneous differential equations

This is more a theoretical question than a specific problem to solve. 

I have just successfully completed a course in differential equations, 
but I really don't understand the difference between homogeneous and 
non-homogeneous equations. First of all, what is the real meaning of a 
homogeneous equation (other than that it equals 0)? Why is the 
solution to the corresponding homogeneous equation to a differential 
equation part of the solution to a non-homogeneous equation? Since the 
homogeneous part of the solution sums to zero, it seems that what is 
being said is that zero plus the particular solution is the complete 
solution. But why can't the zero be ignored, just as 4+2 = 0+6, but 
the zero is ignored? I believe the answer has something to do with 
vector spaces; if that is true, could you explain that in detail?

Thanks very much,
Elliott Zimmermann

Date: 02/26/2001 at 20:22:22
From: Doctor Schwa
Subject: Re: Homogeneous and non-homogeneous differential equations

Hi Elliott,

This does have something to do with *linearity*, though not especially 
with vector spaces.

Remember what our goal is here: not just to find an equation that 
works, like 4 + 2 = 0 + 6, but to find ALL solutions of an equation 
like x^2 + x = 0 + 6.

Now, here's where the linearity is crucial. x = -1 is a solution to 
x^2 + x = 0, and x = 2 is a solution to x^2 + x = 6, but that does NOT 
mean that x = (-1 + 2) will be a solution to x^2 + x = (0 + 6). That's 
because the squaring messes things up; you get that middle term when 
you square a sum.

But if you had a LINEAR equation, like x + y = 0 + 6, you COULD find 
all the solutions by saying: y = -x is ALL the solutions to x + y = 0, 
and (2,4) is A solution to x + y = 6, so (2+x,4-x) gives ALL solutions 
to x + y = 0 + 6 = 6.

In other words, if you "ignore the zero" and just find A solution that 
works like (2,4), you haven't found ALL solutions until you add the 
GENERAL solution that equals 0.

The same exact pattern is true for LINEAR differential equations; if 
you have A solution that works for the non-homogenous case, then 
adding to it ALL solutions that work for the homogeneous equation 
gives you ALL solutions to the non-homogeneous equation.

I hope that helps clear things up. Feel free to write back if you'd 
like to discuss this important topic further.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/