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प्रश्न
Solve the following differential equation:-
\[\frac{dy}{dx} + \left( \sec x \right) y = \tan x\]
उत्तर
We have,
\[\frac{dy}{dx} + \left( \sec x \right)y = \tan x\]
\[\text{Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get}\]
\[P = \sec x\]
\[Q = \tan x\]
Now,
\[I . F . = e^{\int\sec x dx} \]
\[ = e^{\log\left| \left( \sec x + \tan x \right) \right|} \]
\[ = \sec x + \tan x\]
So, the solution is given by
\[y \times I . F = \int Q \times I . F . dx + C\]
\[ \Rightarrow y\left( \sec x + \tan x \right) = \int\left( \sec x + \tan x \right)\tan x + C\]
\[ \Rightarrow y\left( \sec x + \tan x \right) = \int\sec x \times \tan x dx + \int \tan^2 x dx + C\]
\[ \Rightarrow y\left( \sec x + \tan x \right) = \int\sec x \times \tan x dx + \int\left( \sec^2 x - 1 \right) dx + C\]
\[ \Rightarrow y\left( \sec x + \tan x \right) = \sec x + \tan x - x + C\]
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