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Question
For the following differential equation, find a particular solution satisfying the given condition:- \[\frac{dy}{dx} = y \tan x, y = 1\text{ when }x = 0\]
Solution
We have,
\[\frac{dy}{dx} = y \tan x\]
\[ \Rightarrow \frac{1}{y}dy = \tan x dx\]
Integrating both sides, we get
\[\int\frac{1}{y}dy = \int\tan x dx\]
\[ \Rightarrow \log y = \log \left| \sec x \right| + C . . . . . . . \left( 1 \right)\]
Now,
When `x = 0, y = 1`
\[ \therefore \log 1 = \log 1 + C\]
\[ \Rightarrow C = 0\]
Putting the value of `C` in (1), we get
\[\log y = \log \left| \sec x \right|\]
\[ \Rightarrow y = \sec x\]
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