JEE Advanced 2018 · Paper 2 · Mathematics Q10

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MathematicsNumerical+3Question 46 of 54

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function with $f(0) = 0$ . If $y = f(x)$ satisfies the differential equation

$\frac{dy}{dx} = (2 + 5y)(5y - 2),$

then the value of $\displaystyle\lim_{x \to -\infty} f(x)$ is _____.

Your answer

Answer and worked solution

Official answer: 0.4

Equation. $\dfrac{dy}{dx}=25y^2-4$ , with equilibria $y=\pm\tfrac25$ .

Direction. $y(0)=0$ lies between them, where $\dfrac{dy}{dx}<0$ . So $y$ decreases as $x$ increases. Going back to $x\to-\infty$ , $y$ increases towards the upper equilibrium, never crossing it.

Explicitly, $y=\dfrac25\cdot\dfrac{1-e^{20x}}{1+e^{20x}}$ , which tends to $\dfrac25$ .

Answer: 0.4

Topics:Variable separable

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JEE Advanced 2018 · Paper 2, Mathematics question 10