The scaling property: if is a solution of the homogeneous equidimensional ODE with , then so is for any constant .

Proof. By the chain rule, and , so

with , since solves the homogeneous equation.

Solving. Since , the function is an eigenfunction of with eigenvalue . Trying in the homogeneous equation gives

For distinct roots the complementary functions are

Equivalently, substituting turns the ODE into one with constant coefficients,

whose characteristic equation is the same as above. For the degenerate case ,