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The solutions to a homogeneous linear differential equation is a vector space.

Created over 8 years ago, updated 10 days ago

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}}).