For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation $$ \frac{dy}{dx} + 12y = \cos\left(\frac{\pi}{12}x\right), y(0) = 0. $$ Then, which of the following statements is/are TRUE?

(C) (C) There exists a real number $\beta$ such that the line $y = \beta$ intersects the curve $y = y(x)$ at infinitely many points

(A) (A) $y(x)$ is an increasing function

(B) (B) $y(x)$ is a decreasing function

(D) (D) $y(x)$ is a periodic function

Differential Equations — JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 Question with Solution

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

Answer: (C)

Solution

The differential equation is linear with integrating factor

e^{12 x}

.

Solution:

y \cdot e^{12 x} = \int e^{12 x} cos (\frac{π}{12} x) d x

Using integration by parts:

y = \frac{12}{1 2 ^{4} + π ^{2}} [1 2^{2} cos (\frac{π x}{12}) + π sin (\frac{π x}{12}) - 1 2^{2} e^{- 12 x}]

Computing

\frac{d y}{d x}

\frac{d y}{d x} = \frac{12}{1 2 ^{4} + π ^{2}} [- 12 π sin (\frac{π x}{12}) + \frac{π ^{2}}{12} cos (\frac{π x}{12}) + 1 2^{3} e^{- 12 x}]

For

x \leq 0

, the exponential term dominates and

\frac{d y}{d x} > 0

.

For

x > 0

, the derivative can be positive or negative depending on the trigonometric terms.

Therefore,

y (x)

is neither strictly increasing nor strictly decreasing. (A) and (B) are false.

Since the solution contains periodic trigonometric terms plus a decaying exponential, for large

x

y (x)

oscillates around a periodic function. There exists

β

such that

y = β

intersects the curve infinitely many times. (C) is true. ✓

y (x)

is not periodic due to the exponential term. (D) is false.

Therefore, only (C) is correct.

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · One or more correct options

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Differential Equations

Solution

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