\(\frac{\pi}{4}\)
\(\frac{3\pi}{4}\)
\(\frac{\pi}{2}\)
\(\frac{3\pi}{2}\)
To solve the given differential equation and find the limit, we must first analyze the given information:
The differential equation is \((1 + e^{2x})\left(\frac{dy}{dx} + y\right) = 1\), and it passes through the point \((0, \frac{\pi}{2})\). We are tasked to find \(\lim_{{x \to \infty}} e^x y(x)\).
Using substitution, let \(u = e^x\), then \(du = e^x dx\) or \(dx = \frac{du}{u}\), and \(\int \frac{du}{u(1 + u^2)} = \arctan(u) + C_1\)
Therefore, the correct answer is \(\frac{3\pi}{4}\).
\(D.E (1 + e^{2x})\frac{dy}{dx} + y = 1\)
\(⇒\) \(\frac{dy}{dx} + y = \frac{1}{1+e^{2x}}\)
\(\text{I.F.} = e^{\int 1 \,dx} = e^x\)
\(∴\)\(e^x y(x) = \int \frac{e^x}{1 + e^{2x}} \,dx\)
\(⇒\)\(e^x y(x) = \tan^{-1}(e^x) + C\)
\(∵\) It passes through
\((0, \frac{\pi}{2}), \quad C = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}\)
\(∴\) \(\lim_{{x \to \infty}} e^x y(x) = \lim_{{x \to \infty}} \tan^{-1}(e^x) + \frac{\pi}{4}\)
\(= \frac{3\pi}{4}\)
So, the correct option is (B): \(\frac{3\pi}{4}\)
In the given figure, the blocks $A$, $B$ and $C$ weigh $4\,\text{kg}$, $6\,\text{kg}$ and $8\,\text{kg}$ respectively. The coefficient of sliding friction between any two surfaces is $0.5$. The force $\vec{F}$ required to slide the block $C$ with constant speed is ___ N.
(Given: $g = 10\,\text{m s}^{-2}$) 
Method used for separation of mixture of products (B and C) obtained in the following reaction is: 
Which of the following best represents the temperature versus heat supplied graph for water, in the range of \(-20^\circ\text{C}\) to \(120^\circ\text{C}\)? 
A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution.
Read More: Formation of a Differential Equation