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

Question

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

cdquestions Admin · Accepted Answer

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).

Answer

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

Answer

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

Answer

$\exp \left(\frac{1-\pi}{4 \sqrt{2}}\right)$

Answer

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

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 :

The Correct Option is B

Solution and Explanation

Step 1: Standard Form of the Differential Equation

Step 2: Compute the Integrating Factor

Step 3: Solve the Differential Equation

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

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

Conclusion

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