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/
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