Mathematics

Problem

For a non–zero constant

k

, the differential equation

\dfrac{dy}{dx}=k\,y

arises frequently in models of exponential change. a. (i) Show that

\ln|y|=k x+C_1

, where

C_1

is an arbitrary constant. b. (ii) Hence obtain the general solution

y=C e^{k x}

, clearly identifying the arbitrary constant

C

. c. (iii) Prove, by examining the sign of

\dfrac{dy}{dx}

, that every solution curve is increasing when

k>0

and decreasing when

k<0

. State what happens when

k=0

Intuitive explanation

This equation captures the essence of exponential growth or decay: the rate of change of $y$ is always proportional to $y$ itself. The key insight is that solutions have the form $y = Ce^{kx}$ —the function grows (or shrinks) at a rate determined entirely by its current size. Solving it requires separating variables to isolate $y$ and $x$ on opposite sides, then integrating both sides.

Intuitive explanation

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