For a non–zero constant

MATHEMATICS · Multi-part · CBSE · Mathematics · Differential Equations · Differential Equations

Problem

k

, the differential equation

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

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

\ln|y|=k x+C_1

, where

C_1

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

y=C e^{k x}

, clearly identifying the arbitrary constant

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