JEE Advanced 2022 Paper 2, Mathematics Section 2 Q6: Methods of Solving First Order, First Degree Differential Equation

JEE Advanced2022Paper 2Mathematics Section 2

For $x \in R$ , let the function $y (x)$ be the solution of the differential equation

$\frac{d y}{d x} + 12 y = cos (\frac{π}{12} x)$ , $y (0) = 0$ .

Then, which of the following statements is/are TRUE?

A
$y (x)$ is an increasing function
B
$y (x)$ is a decreasing function
C
There exists a real number $β$ such that the line $y = β$ intersects the curve $y = y (x)$ at infinitely many points
D
$y (x)$ is a periodic function

Solution

The integrating factor is $e^{12 x}$ . Solving the linear ODE gives

$y (x) = \frac{1}{144 + π ^{2} /144} [12 cos \frac{π x}{12} + \frac{π}{12} sin \frac{π x}{12}] + C e^{- 12 x}$ ,

with $C$ determined by $y (0) = 0$ . As $x \to \infty$ , the exponential term decays, and $y (x)$ approaches a sinusoidal function with period $24$ .

Hence $y$ is neither monotonic nor periodic globally, ruling out (A), (B), (D).

However, because the asymptotic part is periodic, for any value $β$ strictly between its maximum and minimum, the horizontal line $y = β$ meets the curve infinitely often. So (C) is true.

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JEE Advanced 2022 Paper 2, Mathematics Section 2 Q6: Methods of Solving First Order, First Degree Differential Equation

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