JEE Mathematics 2023 Question with Solution

Given: The differential equation is

$$\frac{dy}{dx} - \frac{3x^5 \tan^{-1}(x^3)}{(1 + x^6)^{3/2}} y = 2x$$

and the solution curve passes through the origin, so $y(0)=0$.

Find: The value of $y(1)$.

From the solution, the integrating factor is taken as

$$\text{I.F.} = e^{\int -\frac{3x^5 \tan^{-1}(x^3)}{(1 + x^6)^{3/2}} \, dx}$$

and then written as

$$\text{I.F.} = e^{\tan^{-1}(x^3) \cdot \frac{x^3}{\sqrt{1 + x^6}}}$$

Using this working, the equation becomes

$$y \cdot e^{\tan^{-1}(x^3) \cdot \frac{x^3}{\sqrt{1 + x^6}}} = \int 2x \, dx + C = x^2 + C$$

Applying the condition $y(0)=0$ gives

$$0 = 0 + C \implies C=0$$

Hence,

$$y \cdot e^{\tan^{-1}(x^3) \cdot \frac{x^3}{\sqrt{1 + x^6}}} = x^2$$

At $x=1$,

$$y(1) \cdot e^{\tan^{-1}(1) \cdot \frac{1}{\sqrt{2}}} = 1$$

Since $\tan^{-1}(1)=\frac{\pi}{4}$,

$$y(1) \cdot e^{\frac{\pi}{4\sqrt{2}}} = 1$$

Therefore,

$$y(1) = e^{-\frac{\pi}{4\sqrt{2}}}$$

the solution then rewrites this as

$$\exp\left(\frac{4 - \pi}{4\sqrt{2}}\right)$$

and explicitly states The Correct Option is B. It also contains conflicting labels later, but by the stated answer on the solution, the correct option is B.

The solution is internally inconsistent:

1. It first says The Correct Option is B.
2. It later writes the boxed value as

$$\boxed{\exp\left(\frac{4 - \pi}{4\sqrt{2}}\right)}$$

3. It finally says Therefore, the correct answer is (2).

Since the solution is the primary source for answer resolution and twice points to the second option, the extracted answer is mapped to B.

Also, the boxed expression matches option A, not option B, so there is a discrepancy between the displayed expression and the declared option label on the solution's.