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Question
`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`
Solution
We have,
`(dy)/(dx)+ y tan x = x^n cos x`
\[\text{Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get}\]
\[P = \tan x \]
\[Q = x^n \cos x\]
Now,
\[I . F . = e^{\int\tan x dx} \]
\[ = e^{\log\left( sec x \right)} \]
\[ = \sec x\]
So, the solution is given by
\[y \times I . F . = \int Q \times I . F . dx + C\]
\[ \Rightarrow y \sec x = \int x^n \cos x \sec x\ dx + C\]
\[ \Rightarrow y \sec x = \int x^n dx + C\]
\[ \Rightarrow y \sec x = \frac{x^{n + 1}}{n + 1} + C\]
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