Question

The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if

Options

• k = 0

• k > 0

• k < 0

• none of these

Answer 1

k < 0

 

We have,

\[ \Rightarrow \frac{dy}{dx} - ky = 0\]

\[ \Rightarrow \frac{dy}{dx} = ky\]

\[ \Rightarrow \frac{1}{y}dy = k dx\]

Integrating both sides, we get

\[\int\frac{1}{y}dy = k\int dx\]

\[ \Rightarrow \log\left| y \right| = kx + C . . . . . \left( 1 \right)\]

Now,

\[y\left( 0 \right) = 1\]

\[ \therefore C = 0\]

\[\text{Putting }C = 0\text{ in }\left( 1 \right),\text{ we get }\]

\[\log\left| y \right| = kx\]

\[ \Rightarrow e^{kx} = y\]

According to the question,

\[ e^{k \infty} = 0\]

\[\text{ Since }e^{- \infty} = 0\]

\[ \therefore k < 0.\]