Linear algebra: What is the difference between homogenous and particular solutions?
First, I would like to mention I'm new to asking questions here, though I have found many answers here! I hope to get more involved here over time, I really like this site. If you have any suggestions or comments, (I think you can do this) please message me or leave a comment!
My question is regarding homogeneous and particular solutions. Perhaps a bit embarrassingly, I must admit I still don't fully understand the difference even though I am on chapter 13 of 17 in this book. I understand the two terms as follows: Homogenous solution - if x + y = b, then any ax + ay = b is also true, for any real number, except perhaps zero (if b is nonzero). Particular solution - any specific solution to the system.
The question from the book:
Suppose that MX=V is a linear system, for some matrix M and some vector V. Let the vector P be a particular solution to the system and the vector H a homogeneous solution to the system. Which of the following vectors must be a particular solution to the system? Select all that apply.
A. P-2H
B. P+2H
C. 2P+2H
D. 2H-P
E. 3H-P
F. 3P+H
G. P-3H
H. 2H-P
I. 2P+H
J. P-H
K. P+H
L. H
M. H-P
N. P+3H
O. P
P. 3P+3HHonestly, I tried a few different combinations, but I really don't know what works. At first, I thought all were true that had only "P" added by any amount of H (such as P - H, P + 3H, and P, but I also included H). For a moment, I thought there only L and O would work, but that didn't provide the correct answer (I do not know the answer, only that I am getting it wrong). Alas, my numerous guesses failed. I suppose it shows I really don't understand the terms. I tried finding the answer online, but most of the results concern either chemistry or differential equations.
Thank you for the help! Please don't hesitate to message me or leave a comment. I am new to participating here, so any suggestions are nice!
$\endgroup$ 12 Answers
$\begingroup$The general equation is $$ A x = b $$
If $b \ne 0$ this is called inhomogenous equation, if $b = 0$ this is called homogeneous equation.
A particular solution $x_p$ is a solution of the inhomogenous equation, thus one of perhaps many solutions. $$ A x_p = b $$
All solutions of the inhomogenous equation can be found by finding all solutions of the homogenous equation and then adding the particular solution. $$ A (x_h + x_p) = A x_h + A x_p = 0 + b = b $$
This is quite useful and is applied from linear differential equations to linear Diophantine equations.
About your multiple choice question: $$ M(P + k H) = M P + k M H = V + k 0 = V $$ Thus select all those solutions with positive one $P$ and arbitrary many $H$. If in doubt apply $M$ to the candidate and check if you get $V$ as result.
$\endgroup$ 1 $\begingroup$The idea behind distinguishing particular (also known as inhomogeneous) and homogeneous solutions is that there's actually a whole linear subspace of such 'homogeneous' solutions $x$ to $Ax=0$ (called the homogeneous problem) known as the kernel or null space of $A$ (as evidenced by the fact that $\alpha x_1+\beta x_2$ is also a solution to $Ax=b$ given solutions $x_1,x_2$).
Particular solutions, on the other hand, cannot form such a subspace; consider solutions $x_1,x_2$ to $Ax=b$. We see then that $A(\alpha x_1+\beta x_2)=\alpha Ax_1+\beta Ax_2=(\alpha+\beta)b\ne b$ in general.
In fact, the set of solutions to $Ax=b$ is not a linear subspace but an affine one -- here, a particular solution $x_p$ to $Ax=b$ (our 'origin' for our affine space) gives rise to displacement vectors $x-x_p$ which then comprise a linear subspace of solutions to the homogeneous problem $Ax=0$.
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