JEE Main 2022 · 28 Jun · Shift 2 · Mathematics Q10

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MathematicsSingle correct+4 / -1Question 10 of 90

Let $x=x(y)$ be the solution of the differential equation $2y\,e^{x/y^2}dx+\left(y^2-4xe^{x/y^2}\right)dy=0$ such that $x(1)=0$ . Then, $x(e)$ is equal to :

e\log_e(2)

-e\log_e(2)

e^2\log_e(2)

-e^2\log_e(2)

Answer and worked solution

Official answer: D

Substitute $x=vy^2$ . $dx=y^2dv+2vy\,dy$ , and the equation reduces to $2y^3e^v\,dv+y^2dy=0$ , i.e. $2e^vdv=-\frac{dy}y$ .

Integrate. $2e^v=-\ln y+C$ ; $x(1)=0$ gives $C=2$ .

At $y=e$ . $2e^v=1$ , $v=-\ln2$ , so $x=-e^2\ln2$ .

(D)

Topics:Homogeneous equations

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JEE Main 2022 · 28 Jun · Shift 2, Mathematics question 10