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 :
- $\exp \left(\frac{4+\pi}{4 \sqrt{2}}\right)$
- $\exp \left(\frac{4-\pi}{4 \sqrt{2}}\right)$
- $\exp \left(\frac{1-\pi}{4 \sqrt{2}}\right)$
- $\exp \left(\frac{\pi-4}{4 \sqrt{2}}\right)$
The Correct Option is B
Solution and Explanation
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).
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