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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
Solution
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,
\[ e^{k \infty} = 0\]
\[\text{ Since }e^{- \infty} = 0\]
\[ \therefore k < 0.\]
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