Two families of functions are proposed as potential general solutions to the…

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

Two families of functions are proposed as potential general solutions to the differential equation

y'' - y' - 2y = 0

. Family A:

y = C_1 e^{2x} + C_2 e^{-x}

Family B:

y = (C_1 + C_2 x) e^{x}

Here

C_1

and

C_2

denote arbitrary real constants. (1) For Family A, verify whether every member satisfies the given differential equation. Show all necessary work. (2) Repeat the verification for Family B. (3) Based on your work in parts (a) and (b), identify which family (if any) represents the complete set of solutions to the differential equation and justify your conclusion.

Related theory

Modeling
Second-Order Equations
Differential Equations

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