Question

If $y=\left(\frac{2}{\mathrm{\pi }}x-1\right)\mathrm{cosec}\left(x\right)$ is the solution of the differential equation, $\frac{dy}{dx}+p\left(x\right)y=\frac{2}{\mathrm{\pi }}\mathrm{cosec}\left(x\right);0<\mathrm{x}<\frac{\mathrm{\pi }}{2}$, then the function $p\left(x\right)$ is equal to

• A. $\mathrm{cosec}\left(x\right)$
• B. $\cot \left(x\right)$
• C. $\tan \left(x\right)$
• D. $\sec \left(x\right)$

Answer 1

The correct option is B

$\cot \left(x\right)$

Explanation for the correct option.

Step 1. Find the differential equation.

Differentiate the equation $y=\left(\frac{2}{\mathrm{\pi }}x-1\right)\mathrm{cosec}\left(x\right)$ with respect to $x$.

$$\begin{gathered} \frac{dy}{dx}=\frac{d}{dx}\left[\left(\frac{2}{\mathrm{\pi }}x-1\right)\mathrm{cosec}\left(x\right)\right] \\ \Rightarrow \frac{dy}{dx}=\frac{2}{\mathrm{\pi }}\mathrm{cosec}\left(x\right)+\left(\frac{2}{\mathrm{\pi }}x-1\right)(-\mathrm{cosec}\left(x\right)\cot \left(x\right)) \\ \Rightarrow \frac{d\mathrm{y}}{d\mathrm{x}}=\frac{2}{\mathrm{\pi }}\mathrm{cosec}\left(\mathrm{x}\right)-\left(\frac{2}{\mathrm{\pi }}\mathrm{x}-1\right)\mathrm{cosec}\left(\mathrm{x}\right)\cot \left(\mathrm{x}\right) \\ \Rightarrow \frac{d\mathrm{y}}{d\mathrm{x}}=\frac{2}{\mathrm{\pi }}\mathrm{cosec}\left(\mathrm{x}\right)-y\cot \left(\mathrm{x}\right)\left[\mathit{y}\mathbf{=}\mathbf{(}\frac{\mathbf{2}}{\mathbf{\pi }}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{cosec}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\right] \\ \Rightarrow \frac{d\mathrm{y}}{d\mathrm{x}}+y\cot \left(\mathrm{x}\right)=\frac{2}{\mathrm{\pi }}\mathrm{cosec}\left(\mathrm{x}\right) \end{gathered}$$

Step 2. Find the function $p\left(x\right)$.

Comparing the given differential equation $\frac{dy}{dx}+p\left(x\right)y=\frac{2}{\mathrm{\pi }}\mathrm{cosec}\left(x\right)$, with the equation found $\frac{d\mathrm{y}}{d\mathrm{x}}+y\cot \left(\mathrm{x}\right)=\frac{2}{\mathrm{\pi }}\mathrm{cosec}\left(\mathrm{x}\right)$ it can be seen that the function $p\left(x\right)$ corresponds to $\cot \left(x\right)$.

So the function $p\left(x\right)$ is equal to $\cot \left(x\right)$.

Hence, the correct option is B.