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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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 fo

Vedclass · Accepted Answer

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

Vedclass · Answer

(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?

Similar Questions

If the line $\vec{r} = (\hat{i} - 2\hat{j} - \hat{k}) + \lambda (2\hat{i} + \hat{j} + 2\hat{k})$ is parallel to the plane $\vec{r} \cdot (3\hat{i} - 2\hat{j} - m\hat{k}) = 14$,then the value of $m$ is:

The angle between the lines whose direction cosines are given by the equations $l^2+m^2-n^2=0$ and $l+m+n=0$ is

Assertion : Yeasts such as $Saccharomyces \text{ } cerevisiae$ are used in baking industry.
Reason : Carbon dioxide produced during fermentation causes bread dough to rise by thermal expansion.

The length of the common chord of the two circles $x^2+y^2-4y=0$ and $x^2+y^2-8x-4y+11=0$ is

The overall goal of glycolysis,Krebs' cycle,and the electron transport system is the formation of

Vedclass Test Series

Exam Paper Generator

Online Exam Module

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?