QJEEAdv22Mae485: JEE Advanced 2022 Mathematics - Class 12 JEEAdv22Mae485 Solution: For , let the function be the solution of the differential…

Question

Verified Solution • 2025–26 Session moderate

For

x \in \mathbb{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), \quad 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 $\beta$ such that the line $y=\beta \quad$ intersects the curve $y=y(x)$ at infinitely many points
D $y(x)$ is a periodic function

Accepted Answer

Given,

\frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right)

This is a linear differential equation.

Where P = 12 and Q =

\cos \left(\frac{\pi}{12} x\right)

\text { I.F. }=e^{\int P d x}=e^{\int 12 d x}=e^{12 x}

Solution of differencial equation is

y \times I.F = \int {Q \times \left( {I.F} \right)} dx

y \times \text { I.F. }=\int e^{12 x} \cdot \cos \left(\frac{\pi}{12} x\right) d x

\Rightarrow y e^{12 x}=\frac{e^{12 x}}{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

[As

\int {{e^{ax}}.\cos \left( {bx} \right)dx} = {{{e^{ax}}} \over {{a^2} + {b^2}}}\left\{ {a\cos \left( {bx} \right) + b\sin \left( {bx} \right)} \right\} + C

]

\Rightarrow y=\frac{12}{(12)^4+\pi^2}\left\{(12)^2 \cos \left(\frac{\pi x}{12}\right)+\pi \sin \left(\frac{\pi x}{12}\right)\right\}+\frac{C}{e^{12 x}}

\begin{aligned}&amp; \text { Given }, y(0)=0 \\\\&amp; \Rightarrow 0=\frac{12 \cdot\left(12^2\right)}{12^4+\pi^2}+C \\\\&amp; \Rightarrow C=\frac{-12^3}{12^4+\pi^2} \\\\&amp; \therefore y=\frac{12}{12^4+\pi^2}\left[12^2 \cos \left(\frac{\pi x}{12}\right)+\pi \sin \left(\frac{\pi x}{12}\right)-12^2 \cdot e^{-12 x}\right]\end{aligned}

y(x)

is not a periodic function as

e^{-12 x}

is a non-periodic function.

\therefore

Option (D) is a wrong statement.

\begin{aligned}&amp; \text { Now } \frac{d y}{d x}=\frac{12}{12^4+\pi^2}[{-12 \pi \sin \left(\frac{\pi x}{12}\right)+\frac{\pi^2}{12} \cos \left(\frac{\pi x}{12}\right)}+12^3 \mathrm{e}^{-12 x}]\end{aligned}

Values of

{-12 \pi \sin \left(\frac{\pi x}{12}\right)+\frac{\pi^2}{12} \cos \left(\frac{\pi x}{12}\right)}

varies between

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

So,

\mathrm{f}(\mathrm{x})

is neither increasing nor decreasing.

\therefore

Option (A) and (B) are wrong statements.

For some

\beta \in R, y=\beta

intersects

y=f(x)

at infinitely many points.

So option

\mathrm{C}

is correct.

Step-by-Step Verified Solution