Consider the slope field for the differential equation

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

Problem

\dfrac{dy}{dx}=3x^{2}

. The analytic family of solutions is

y=x^{3}+C

. Which statement best explains why specifying the point

(1,4)

selects exactly one integral curve from this family? (a) Knowing

y(1)=4

uniquely determines the slope

\dfrac{dy}{dx}

x=1

, which in turn fixes the entire curve. (b) Substituting

x=1

into

\dfrac{dy}{dx}=3x^{2}

shows the slope there is

3

, so integrating

y'=3

gives

y=3x+C

; passing through

(1,4)

sets

C=1

, producing a single curve. (c) Substituting

x=1

and

y=4

into

y=x^{3}+C

determines a single value of

C

, so only one curve satisfies the condition. (d) Differentiating

y=x^{3}+C

gives

\dfrac{dy}{dx}=3x^{2}

, which forces

C=0

. Therefore

(1,4)

selects the unique curve

y=x^{3}