$y(x)$ is an increasing function.
$y(x)$ is a decreasing function.
There exists a real number $\beta$ such that the line $y =\beta$ intersects the curve $y = y ( x )$ at infinitely many points.
$y ( x )$ is a periodic function.
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.
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$, then the value of $\alpha$ is ____
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Differential equations can be divided into several types namely