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Problem 71: $V = C^2(I)$, and S is the subset of V consisting of those functions satisfying the differential equation $y"+2y'-y$ on I

Problem 71: $V = C^2(I)$, and S is the subset of V consisting of those functions satisfying the differential equation $y''+2y'-y=1$ on I.

Solution:
$S = \{y \in C^2(I):y''+2y'-y=1\}$.
Closure under Addition:
Let $y_1, y_2 \in S$.
$(y_1+y_2)''+2(y_1+y_2)'-(y_1+y_2) = y_1''+y_2''+2y_1'+2y_2'-y_1-y_2$
$= (y_1''+2y_1'-y_1)+(y_2''+2y_2'-y_2) = 1 +1 = 2 \ne 1$
Hence,S is not closed under addition.
Closure under Scalar Multiplication:
Let $k \in \R$ and $y_1 \in S$
$(ky)''+2(ky)'-(ky) = ky''+2ky'-ky= k(y''+2y'-y) = k1= k \ne 1$
which shows that $ky_1 \notin S$
Therefore S is NOT a subspace of $C^2(I)$

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