Complex solutions to a differential equation a vector space?

[csnsc14320]

Homework Statement

Is the set of all complex solutions to the differential equation $$\frac{d^2 y}{d x^2} + 2\frac{d y}{d x} - 3 y = 0$$

If so, find a basis, the dimension, and give the zero vector

Homework Equations

The Attempt at a Solution

I solved the equation and got the answer:

$$y(x) = C_1 e^{-3x} + C_2e^x$$

I know how to test if a set is a vector space but I'm not really seeing the "set" here. Is it because $$C_1$$ and $$C_2$$ can be complex numbers? In which case, wouldn't any complex number work so would I get the set of all complex numbers?

any help is appreciated

 

[lanedance]

i think you missed out somthing in your question -

I'm assuming it is, "is the set of all complex solutions to the differential equation - a vector space"

I would start with the axioms for a vector space - what are they?
Then what is the general form of your solution? This will generally have some undetermined constants to give a family of solutions

"the space" is then the set of all solutions. Is it a vector space?

In short, a vector space is closed under scalar multiplication and addition, with some other axioms, so check:
closure under scalar multiplication - so given any solution is a scalar times the solution also a solution
closure under addition - so given 2 solutions is their sum also a solution

then fill out the other axioms

 

[lanedance]

updated post above

 

[HallsofIvy]

Homework Statement

Is the set of all complex solutions to the differential equation $$\frac{d^2 y}{d x^2} + 2\frac{d y}{d x} - 3 y = 0$$

If so, find a basis, the dimension, and give the zero vector

Homework Equations

The Attempt at a Solution

I solved the equation and got the answer:

$$y(x) = C_1 e^{-3x} + C_2e^x$$

I know how to test if a set is a vector space but I'm not really seeing the "set" here. Is it because $$C_1$$ and $$C_2$$ can be complex numbers? In which case, wouldn't any complex number work so would I get the set of all complex numbers?

any help is appreciated

First, exactly what do you mean by "complex solution"? If you mean simply that the coefficients C₁ and C₂ are complex, as you say, they the "set" asked about is NOT "the set of all complex numbers". It is the set of all such functions:
$$\{ f(x)= C_1 e^{-3x}+ C_2 e^x : C_1, C_2 \in \math{C}\}$$.

If you add two such functions is the sum also a function of that kind? If you multiply such a function by a complex number is the product also a function of that kind?