JEE 2023 — Mathematics PYQ

1. Identify the Type of Differential Equation

The given equation is a First-Order Linear Differential Equation of the form:

By comparing, we have:

• $P(x)=\tan x$

• $Q(x)=x\sec x$

2. Find the Integrating Factor (I.F.)

The formula for the Integrating Factor is $I.F.=e^{\int P(x)dx}$:

3. General Solution

The solution is given by:

Applying Integration by Parts ($\int uvdx=u\int vdx-\int (u^{\prime }\int vdx)dx$):

Let $u=x$ and $v={\sec }^{2}x$.

4. Apply Initial Condition

Given $y(0)=1$, substitute $x=0$ and $y=1$:

So, the particular solution is:

5. Calculate $y\left(\frac{\pi }{6}\right)$

Substitute $x=\frac{\pi }{6}$ into the equation:

Multiply both sides by $\frac{\sqrt{3}}{2}$:

Since $1=\ln (e)$:

To match the options, we can invert the log fraction by changing the sign:

Correct Option: (2)