Question

`(dy)/(dx)+ y tan x = x^n cos x, n ne− 1`

Answer 1

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

\[y \times I . F . = \int Q \times I . F . dx + C\]

\[ \Rightarrow y \sec x = \int x^n \cos x \sec x\ dx + C\]

\[ \Rightarrow y \sec x = \int x^n dx + C\]

\[ \Rightarrow y \sec x = \frac{x^{n + 1}}{n + 1} + C\]