Question

Let the solution curve $y=y(x)$ of the differential equation $\frac{d y}{d x}-\frac{3 x^5 \tan ^{-1}\left(x^3\right)}{\left(1+x^6\right)^{3 / 2}} y=2 x \exp \left\{\frac{x^3-\tan ^{-1} x^3}{\sqrt{\left(1+x^6\right)}}\right\}$ pass through the origin Then $y(1)$ is equal to :

• A. $\exp \left(\frac{4+\pi}{4 \sqrt{2}}\right)$
• B. $\exp \left(\frac{4-\pi}{4 \sqrt{2}}\right)$
• C. $\exp \left(\frac{1-\pi}{4 \sqrt{2}}\right)$
• D. $\exp \left(\frac{\pi-4}{4 \sqrt{2}}\right)$

Answer 1

The Correct Option is B

Step 1: Standard Form of the Differential Equation

The given differential equation is:

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

The integrating factor (\(\text{I.F.}\)) is given by:

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

Step 2: Compute the Integrating Factor

The integral simplifies as:

\[ \int -\frac{3x^5 \tan^{-1}(x^3)}{(1 + x^6)^{3/2}} dx = \tan^{-1}(x^3) \cdot \frac{x^3}{\sqrt{1 + x^6}}. \]

Thus, the integrating factor becomes:

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

Step 3: Solve the Differential Equation

The general solution of the differential equation is:

\[ y \cdot \text{I.F.} = \int 2x \cdot \text{I.F.} dx + C. \]

Substitute the I.F.:

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

Simplify:

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

Step 4: Apply the Condition (Passes Through the Origin)

At \(x = 0\), \(y = 0\). Substitute:

\[ 0 \cdot e^{\tan^{-1}(0) \cdot 0 \cdot \sqrt{1 + 0^6}} = 0^2 + C \implies C = 0. \]

Thus, the solution becomes:

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

Step 5: Evaluate \(y(1)\)

At \(x = 1\):

\[ y(1) \cdot e^{\tan^{-1}(1^3) \cdot \frac{1^3}{\sqrt{1 + 1^6}}} = 1^2. \]

Simplify the terms:

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

Rewriting the exponent:

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

Express the exponent further:

\[ y(1) = \exp\left(\frac{4 - \pi}{4\sqrt{2}}\right). \]

Conclusion

The value of \(y(1)\) is:

\[ \boxed{\exp\left(\frac{4 - \pi}{4\sqrt{2}}\right)}. \]

Therefore, the correct answer is (2).