Observe the following statements:I. If dy + 2 | vedclass

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(C) $I$. Given $dy + 2xy dx = 2e^{-x^2} dx$.Dividing by $dx$, we get $\frac{dy}{dx} + 2xy = 2e^{-x^2}$.This is a linear differential equation of the f

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$I$ is true, but $II$ is false

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(C) $I$. Given $dy + 2xy dx = 2e^{-x^2} dx$.Dividing by $dx$, we get $\frac{dy}{dx} + 2xy = 2e^{-x^2}$.This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$, where $P = 2x$ and $Q = 2e^{-x^2}$.The Integrating Factor $(IF)$ is $e^{\int P dx} = e^{\int 2x dx} = e^{x^2}$.The solution is $y(IF) = \int Q(IF) dx + c$.$y e^{x^2} = \int 2e^{-x^2} \cdot e^{x^2} dx + c = \int 2 dx + c = 2x + c$.Thus, statement $I$ is true.$II$. Given $y e^{-x^2} - 2x = c$.Differentiating with respect to $x$, we get $\frac{d}{dx}(y e^{-x^2}) - 2 = 0$.Using the product rule: $e^{-x^2} \frac{dy}{dx} + y e^{-x^2}(-2x) - 2 = 0$.$e^{-x^2} \frac{dy}{dx} = 2 + 2xy e^{-x^2}$.Multiplying by $e^{x^2}$, we get $\frac{dy}{dx} = 2e^{x^2} + 2xy$.This does not match the given expression $dx = (2e^{-x^2} - 2xy) dy$. Thus, statement $II$ is false.

Observe the following statements:
$I$. If $dy + 2xy dx = 2e^{-x^2} dx$, then $ye^{x^2} = 2x + c$.
$II$. If $ye^{-x^2} - 2x = c$, then $dx = (2e^{-x^2} - 2xy) dy$.
Which of the following is a correct statement?

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Observe the following statements:$I$. If $dy + 2xy dx = 2e^{-x^2} dx$, then $ye^{x^2} = 2x + c$.$II$. If $ye^{-x^2} - 2x = c$, then $dx = (2e^{-x^2} - 2xy) dy$.Which of the following is a correct statement?