I'm stuck on my homework on "Definition of A Vector Space" in my Differential Equations and Linear Algebra class. The section of problems I'm on ask you to determine whether or not each problem is closed under addition and/or scalar multiplication. I believe that I successfully did that for:
1) the set of all rational numbers and
2) the set of all *n* x *n* matrices with real elements.
Now that I've done that, I have two linear differential equations to determine the two closures for. I'm not sure how to begin with these and I've searched everywhere. These are the two equations that I can't figure out.
1) The set of all solutions to: y'+9y=4x^2
2) The set of all solutions to: y'+9y=0
It specifies that I cannot solve the differential equation. Thanks for the help in advance guys.
Best Answer
Hint: suppose you had solutions $y_1$ and $y_2$ that satisfies the differential equations. To show closure under addition, you must show that $y_1 + y_2$ also satisfies the equation and that $c y_1$ does as well, where $c$ is a real constant. You will need to use the fact that $y_1$ and $y_2$ are known to satisfy the ODEs.
$\textbf{EDIT:}$ Let $y_1, y_2$ be solutions to $y'+9y= 0$. Then $$(y_1+y_2)' + 9 (y_1+y_2) = y_1' + y_2' + 9y_1 + 9y_2 = (y_1' + 9y_1) + (y_2' + 9y_2) = 0 + 0 = 0.$$ We have shown closure under addition for this particular case.