Question

For $x \in R$, let the function $y ( x )$ be the solution of the differential equation $\frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), y(0)=0 \text { }$.
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 $\beta$ such that the line $y =\beta$ intersects the curve $y = y ( x )$ at infinitely many points.
• D. $y ( x )$ is a periodic function.

Answer 1

The Correct Option is C

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

Answer 2

Given,
\(\frac{dy}{dx} + 12y = \cos \left( \frac{\pi}{12} x \right)\)

This is a linear differential equation. 
 \(\text{I.F.} = e^{\int 12 \, dx} = e^{12x}\)

The solution of the differential equation becomes:
\(y \cdot e^{12x} = \int e^{12x} \cdot \cos \left( \frac{\pi}{12} x \right) \, dx\)

\(y \cdot e^{12x} = \frac{e^{12x}}{(12)^2 + \left( \frac{\pi}{12} \right)^2} \left( 12 \cos \frac{\pi}{12} x + \frac{\pi}{12} \sin \frac{\pi}{12} x \right) + C\)

\(\Rightarrow y = \frac{12}{(12)^2 + \pi^2} \left( (12)^2 \cos \left( \frac{\pi}{12} x \right) + \pi \sin \left( \frac{\pi}{12} x \right) \right) + \frac{C}{e^{12x}}\)

Given that y(0) = 0:
\(0 = \frac{12}{124 + \pi^2} \left( 12^2 \cdot 0 + \pi \cdot 0 \right) + C \Rightarrow C = \frac{-12^3}{124 + \pi^2}\)

\(\therefore y = \frac{12}{124 + \pi^2} \left( (12)^2 \cos \left( \frac{\pi}{12} x \right) + \pi \sin \left( \frac{\pi}{12} x \right) \right) - 12^2 \cdot e^{-12x}\)

Now, finding \(\frac{dy}{dx}\):
\(\frac{dy}{dx} = \frac{12}{124 + \pi^2} \left[ -12 \pi \sin \left( \frac{\pi}{12} x \right) + \pi^2 \cos \left( \frac{\pi}{12} x \right) \right] + 12^3 e^{-12x}\)

\(\left( - \sqrt{144 \pi^2 + \frac{\pi^4}{144}} = -12 \pi \sqrt{1 + \frac{\pi^2}{124}} \right)\)

\(\Rightarrow \frac{dy}{dx} > 0 \text{ for } x \leq 0 \text{ and maybe negative/positive for } x > 0\)

So, f(x) is neither increasing nor decreasing.
For some \(\beta \in  R,y=\beta\) intersects y=f(x) at infinitely many points

So, the correct option is (C): There exists a real number \(\beta\) such that the line \(y=\beta\) intersects the curve y=y(x) at infinitely many points.