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--- math121a-f23:october_27_friday [2023/10/26 22:17]
pzhou created
+++ math121a-f23:october_27_friday [2023/10/26 22:33] (current)
pzhou [Inhomogenous term]
@@ Line 39: / Line 39: @@
 ===== Inhomogenous term =====
-what if you had one
+If you had a equation of the form
+ $D f(x) = 1$
+then it is not a homogeneous equation: the term on the right hand side is does not contain factor  $f$ . What does its solution space look like? We know it is of the form
+ $f(x) = c + x.$
+Note that, the solution space is not a vector space anymore. indeed, if you have  $f_1(x)$  and  $f_2(x)$  both satisfies the equation,
+ $D f_1(x) = 1$
+ $D f_2(x) = 1$
+then add the two equation up, we see
+ $D (f_1(x) + f_2(x)) = 2$
+so  $f_1(x) + f_2(x)$  is not a solution ( $1 \neq 2$ ).
+Nonetheless, the solution space is a so called 'affine space'  $V$ , which means there is an associated vector space  $V'$ , and for any two elements  $v_1, v_2 \in V$ , we have their difference  $v_1 - v_2 \in V'$ .
+In our case, the associated vector space is the solution space for the homogenous equation
+ $V' = \{f(x) \mid Df = 0 \}$
+That means, if we pick a 'base point'  $v_0 \in V$ , and pick a basis  $e_1, \cdots, e_n$  of  $V'$ , and then we can express any element  $v \in V$  as
+ $v = v_0 + (c_1 e_1 + \cdots + c_n e_n).$
+for some coefficients  $c_i$ .
+Back to our problem here, any solution to the inhomogenous equation can be written as a 'particular solution' (playing the role of  $v_0$  above), and plus a solution to the homogenous equation (an element of $V'$).

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