If the solution y(x) of the differential equa | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The given differential equation is $\sin x \frac{dy}{dx} + y \cos x = e^{2x}$.Dividing by $\sin x$, we get $\frac{dy}{dx} + y \cot x = \frac{e^{2x

Vedclass · Accepted Answer

$e^{\pi/3} - e^\pi$

Vedclass · Answer

(B) The given differential equation is $\sin x \frac{dy}{dx} + y \cos x = e^{2x}$.Dividing by $\sin x$, we get $\frac{dy}{dx} + y \cot x = \frac{e^{2x}}{\sin x}$.This is a linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x) = \cot x$ and $Q(x) = \frac{e^{2x}}{\sin x}$.The Integrating Factor ($I$.$F$.) is $e^{\int P(x) dx} = e^{\int \cot x dx} = e^{\ln(\sin x)} = \sin x$.Multiplying the equation by the $I$.$F$.,we get $\frac{d}{dx}(y \sin x) = e^{2x}$.Integrating both sides with respect to $x$, we get $y \sin x = \int e^{2x} dx = \frac{e^{2x}}{2} + C$.Thus, $y = \frac{e^{2x}}{2 \sin x} + \frac{C}{\sin x}$.Using the condition $y\left(\frac{\pi}{2}\right) = 0$, we have $0 = \frac{e^\pi}{2(1)} + \frac{C}{1}$, which gives $C = -\frac{e^\pi}{2}$.Substituting $C$ back, $y = \frac{e^{2x} - e^\pi}{2 \sin x}$.Now, for $x = \frac{\pi}{6}$, $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.Therefore, $y\left(\frac{\pi}{6}\right) = \frac{e^{\pi/3} - e^\pi}{2(1/2)} = e^{\pi/3} - e^\pi$.

If the solution $y(x)$ of the differential equation $\sin x \frac{dy}{dx} + y \cos x = e^{2x}, x \in (0, \pi)$ satisfies $y\left(\frac{\pi}{2}\right) = 0$, then $y\left(\frac{\pi}{6}\right) = $

Similar Questions

Find a particular solution satisfying the given condition: $\frac{dy}{dx} - 3y \cot x = \sin 2x$; $y = 2$ when $x = \frac{\pi}{2}$.

The solution of $(x+y+1) \frac{dy}{dx} = 1$ is

Find a particular solution of the differential equation $\frac{dy}{dx} + y \cot x = 4x \csc x$ $(x \neq 0),$ given that $y=0$ when $x=\frac{\pi}{2}$.

The solution of the differential equation $\frac{dy}{dx} + y = \cos x$ is

The general solution of the differential equation $100 \frac{d^2 y}{dx^2}-20 \frac{dy}{dx}+y=0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module