(a) Give a convincing demonstration that the second-order equation ay'' + by' + cy =­ 0, a, b, and c constants, always possesses at least one solution of the form y₁ = e^m1x, m₁ a constant.

(b) Explain why the differential equation in part (a) must then have a second solution either of the form y₂ = e^m2x or of the form y₂ = xe^m1x, m₁ and m₂ constants.

(c) Can you explain why the statements in parts (a) and (b) above are not contradicted by the answers to Problems 3–5?