A student wants to verify that the function is a solution of the differential equation

MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations

Problem

A student wants to verify that the function

y = x\sqrt{1 + x^{2}}

is a solution of the differential equation

x\,\dfrac{dy}{dx} = \sqrt{1 + x^{2}} + \dfrac{x^{2}}{\sqrt{1 + x^{2}}}

. Which of the following steps is both necessary and sufficient to complete this verification? (a) Integrate the right–hand side of the differential equation and compare the antiderivative with

y

(ignoring the constant of integration). (b) Verify the equation only at

x=0

; if both sides agree there, the function must satisfy the differential equation everywhere. (c) Differentiate

y

with respect to

x

, substitute

\dfrac{dy}{dx}

into the differential equation, and confirm that the resulting identity holds for all

x

in the domain. (d) Compute

\dfrac{dy}{dx}

and set it equal to

\sqrt{1+x^{2}}+\dfrac{x^{2}}{\sqrt{1+x^{2}}}

, since matching the slopes is sufficient.

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