Example: The general solution to (y'' + 3y' + 2y = 0) is (y = c_1 e^{-x} + c_2 e^{-2x}), forming a 2-dimensional vector space with basis ({e^{-x}, e^{-2x}}).
**Theorem:** The set of solutions to a homogeneous linear differential equation is a vector space.
For example, the solutions to \(y'' + 3y' + 2y = 0\) form a vector space under pointwise addition and scalar multiplication.
**Proof:** If \(y_1\) and \(y_2\) are solutions and \(c\) is a scalar:
- \((y_1 + y_2)'' + 3(y_1 + y_2)' + 2(y_1 + y_2) = (y_1'' + 3y_1' + 2y_1) + (y_2'' + 3y_2' + 2y_2) = 0 + 0 = 0\)
- \((cy_1)'' + 3(cy_1)' + 2(cy_1) = c(y_1'' + 3y_1' + 2y_1) = c \cdot 0 = 0\)
**Example:** The general solution to \(y'' + 3y' + 2y = 0\) is \(y = c_1 e^{-x} + c_2 e^{-2x}\), forming a 2-dimensional vector space with basis \(\{e^{-x}, e^{-2x}\}\).