JEE Main 2022 · 26 Jun · Shift 2 · Mathematics Q10

← JEE Main 2022 · 26 Jun · Shift 2

MathematicsSingle correct+4 / -1Question 10 of 90

If the solution of the differential equation
$\frac{dy}{dx}+e^x\left(x^2-2\right)y=\left(x^2-2x\right)\left(x^2-2\right)e^{2x}$
satisfies $y(0)=0$ , then the value of $y(2)$ is ______

-1

e

Answer and worked solution

Official answer: C

Integrating factor. With $u=e^x(x^2-2x)$ , $u'=e^x(x^2-2)$ , so the IF is $e^u$ and the right side is $uu'$ .

Integrate. $ye^u=\displaystyle\int uu'e^u\,dx=e^u(u-1)+C$ , so $y=u-1+Ce^{-u}$ ; $y(0)=0$ ( $u=0$ ) gives $C=1$ .

At $x=2$ . $u=0$ , so $y(2)=-1+1=0$ .

(C)

Topics:Linear first-order equations

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JEE Main 2022 · 26 Jun · Shift 2, Mathematics question 10