A student claims that to verify whether satisfies the differential equation

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

Problem

A student claims that to verify whether

y=x^{2}

satisfies the differential equation

y'' + 3y = 2

, it is sufficient to substitute

y

into the equation without computing any derivatives. Which of the following best evaluates the student's claim? (a) The claim is correct because

y''=2

is constant, so substituting only

y

suffices to check whether the constant

2

on the right side is obtained. (b) The claim is incorrect because the equation involves

y'

, so the first derivative, not the second derivative, must be computed. (c) The claim is incorrect because the equation involves

y''

, so both

y

and

y''

must be substituted. (d) The claim is incorrect because

y''

for

y=x^{2}

0

, leaving

3y

on the left, which cannot equal

2

for all

x