A differentiable function is proposed as a solution to the first-order differential…

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

Problem

A differentiable function

y=g(x)

is proposed as a solution to the first-order differential equation

\frac{dy}{dx}=y^{2}-4.

Without performing any explicit derivative calculations, address the conceptual requirements for verifying whether

g

actually satisfies the differential equation. (a) State precisely what equality must hold between expressions involving

g

and its derivative in order for

g

to qualify as a solution. (b) Explain why substituting

y=g(x)

and

\dfrac{dy}{dx}=g'(x)

into the differential equation provides both a necessary and sufficient test for verification. (c) Describe two distinct ways in which the domain of

g

could affect the validity of the verification process for this differential equation.