Mathematics

Mathematics

Problem

Consider the one-parameter family of differential equations dydx=ay−2\dfrac{dy}{dx}=ay-2, where aa is a real constant. For which value of aa will the slope field contain horizontal segments whenever y=1y=1? a) a=−2a=-2 b) a=3a=3 c) a=2a=2 d) a=0a=0
Intuitive explanation

The slope field shows horizontal segments where the derivative equals zero—that's where the direction arrows point purely left or right. You need to find which value of aa makes dydx=0\frac{dy}{dx}=0 when y=1y=1. Substitute y=1y=1 into the differential equation and solve for aa; this single constraint uniquely determines the parameter.

Intuitive explanation

The slope field shows horizontal segments where the derivative equals zero—that's where the direction arrows point purely left or right. You need to find which value of aa makes dydx=0\frac{dy}{dx}=0 when y=1y=1. Substitute y=1y=1 into the differential equation and solve for aa; this single constraint uniquely determines the parameter.

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