Mathematics

Problem

In a slope field it is observed that, for every vertical line

x=k

, the slope at any point on that line equals

k^{2}

. Which differential equation below is consistent with this observation? (I)

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

(II)

\dfrac{dy}{dx}=y^{2}

(III)

\dfrac{dy}{dx}=xy

Write the Roman numeral of the correct choice.

Intuitive explanation

A slope field shows how the slope $\frac{dy}{dx}$ varies across the plane. The key insight is that if the slope depends only on the $x$ -coordinate (not on $y$ ), then every point sharing the same $x$ -value must have the same slope. Since you're told that on the vertical line $x=k$ , all slopes equal $k^2$ , you need a differential equation where the right side is a function of $x$ alone.

Intuitive explanation

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