Question

The solution of the differential equation $\frac{dy}{dx}-ky=0,y\left(0\right)=1$ approaches to zero when x → ∞, if
(a) k = 0
(b) k > 0
(c) k < 0
(d) none of these

Answer 1

(c) k < 0


$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \Rightarrow \frac{dy}{dx}-ky=0 \\ \Rightarrow \frac{dy}{dx}=ky \\ \Rightarrow \frac{1}{y}dy=kdx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int \frac{1}{y}dy=k\int dx \\ \Rightarrow \log \left|y\right|=kx+C.....\left(1\right) \\ \mathrm{Now}, \\ y\left(0\right)=1 \\ \therefore C=0 \\ \mathrm{Putting}C=0\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ \log \left|y\right|=kx \\ \Rightarrow e^{kx}=y \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}, \\ {e}^{k\infty }=0 \\ \mathrm{Since}{e}^{-\infty }=0 \\ \therefore k<0. \end{gathered}$$