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### Non-homogeneous Differential Equation Solutions

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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
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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/
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