IIT-JEE 2011 · Paper 2 · Mathematics Q55

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MathematicsInteger+4Question 55 of 60

Let $y'(x)+y(x)g'(x)=g(x)g'(x)$ , $y(0)=0$ , $x\in\mathbb R$ , where $f'(x)$ denotes $\dfrac{df(x)}{dx}$ and $g(x)$ is a given non-constant differentiable function on $\mathbb R$ with $g(0)=g(2)=0$ . Then the value of $y(2)$ is

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Linear equation with integrating factor $e^{g(x)}$ :
$\frac d{dx}\left(ye^{g}\right)=gg'e^{g}\implies ye^{g}=(g-1)e^{g}+C.$

Constant. $y(0)=0$ and $g(0)=0$ give $C=1$ , so $y=g-1+e^{-g}$ .

Value. $y(2)=0-1+1=0$ .

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Topics:Linear first-order equations

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IIT-JEE 2011 · Paper 2, Mathematics question 55