$$\begin{gathered} \text{If}\left(\sec x\right)\frac{dy}{dx}-2y=2+3\mathrm{sin }x\text{ and} y(0)=\frac{-}{7}\text{then}y\left(\frac{\pi }{6}\right)\text{is} \\ \end{gathered}$$

The correct answer is -5/2.

Step-by-step Explanation:

We are given the linear differential equation:

\sec x\frac{dy}{dx}-2y=2+3\sin x

First, let's rewrite the differential equation in standard linear form, \frac{dy}{dx}+P\left(x\right)y=Q\left(x\right), by multiplying the entire equation by \cos x:

\frac{dy}{dx}-2\cos x·y=2\cos x+3\sin x\cos x

Here, P\left(x\right)=-2\cos x and Q\left(x\right)=2\cos x+3\sin x\cos x.

Step 1: Find the Integrating Factor (I.F.)

\text{I.F.}=e^{\int P\left(x\right)dx}=e^{\int -2\cos xdx}=e^{-2\sin x}

Step 2: General Solution of the Differential Equation

The general solution is given by:

y·\left(\text{I.F.}\right)=\int Q\left(x\right)·\left(\text{I.F.}\right)dx+C

y{e}^{-2\sin x}=\int (2\cos x+3\sin x\cos x)e^{-2\sin x}dx+C

To evaluate the integral, substitute t=\sin x, which gives dt=\cos xdx:

I=\int (2+3t)e^{-2t}dt

Using integration by parts:

I=(2+3t)\frac{e^{-2t}}{-2}-\int 3\frac{e^{-2t}}{-2}dt

I=-\frac{1}{2}(2+3t)e^{-2t}-\frac{3}{4}{e}^{-2t}

I=-e^{-2t}(1+\frac{3}{2}t+\frac{3}{4})=-e^{-2t}(\frac{7}{4}+\frac{3}{2}t)

Substituting t=\sin x back into the solution:

y{e}^{-2\sin x}=-e^{-2\sin x}(\frac{7}{4}+\frac{3}{2}\sin x)+C

y=-(\frac{7}{4}+\frac{3}{2}\sin x)+C{e}^{2\sin x}

Step 3: Find the constant C using initial condition

Given that y\left(0\right)=-\frac{7}{4}:

-\frac{7}{4}=-(\frac{7}{4}+\frac{3}{2}\sin 0)+C{e}^{2\sin 0}

-\frac{7}{4}=-\frac{7}{4}+C\Rightarrow C=0

So, the particular solution is:

y\left(x\right)=-(\frac{7}{4}+\frac{3}{2}\sin x)

Step 4: Calculate y\left(\frac{\pi }{6}\right)

Substitute x=\frac{\pi }{6}:

y\left(\frac{\pi }{6}\right)=-(\frac{7}{4}+\frac{3}{2}\sin \frac{\pi }{6})

Since \sin \frac{\pi }{6}=\frac{1}{2}:

y\left(\frac{\pi }{6}\right)=-(\frac{7}{4}+\frac{3}{2}·\frac{1}{2})=-(\frac{7}{4}+\frac{3}{4})=-\frac{10}{4}=-\frac{5}{2}

Thus, the value of y\left(\frac{\pi }{6}\right) is -5/2.