If x frac{dy}{dx} + y = x frac{f(xy)}{f'(xy)} | vedclass

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

Question

(A) Given the differential equation: $x \frac{dy}{dx} + y = x \frac{f(xy)}{f'(xy)}$.We know that $\frac{d}{dx}(xy) = x \frac{dy}{dx} + y$.Substituting

Vedclass · Accepted Answer

$Ce^{x^2/2}$

Vedclass · Answer

(A) Given the differential equation: $x \frac{dy}{dx} + y = x \frac{f(xy)}{f'(xy)}$.We know that $\frac{d}{dx}(xy) = x \frac{dy}{dx} + y$.Substituting this into the equation, we get: $\frac{d(xy)}{dx} = x \frac{f(xy)}{f'(xy)}$.Rearranging the terms to separate the variables, we have: $\frac{f'(xy)}{f(xy)} d(xy) = x dx$.Integrating both sides: $\int \frac{f'(xy)}{f(xy)} d(xy) = \int x dx$.This yields: $\ln |f(xy)| = \frac{x^2}{2} + k$, where $k$ is the constant of integration.Taking the exponential of both sides: $|f(xy)| = e^{\frac{x^2}{2} + k} = e^k \cdot e^{\frac{x^2}{2}}$.Letting $C = e^k$, we get: $|f(xy)| = Ce^{\frac{x^2}{2}}$.

If $x \frac{dy}{dx} + y = x \frac{f(xy)}{f'(xy)}$, then $|f(xy)|$ is equal to

Similar Questions

If $\frac{dy}{dx} + \frac{2^{x-y}(2^y - 1)}{2^x - 1} = 0$,$x, y > 0$,and $y(1) = 1$,then $y(2)$ is equal to

The solution of the equation $(1 + x^2)\frac{dy}{dx} = 1$ is

The solution of the differential equation $x\frac{dy}{dx} + y = y^2$ is

The general solution of the differential equation $\log_{e}\left(\frac{dy}{dx}\right) = x + y$ is

The solution of the differential equation $x \frac{dy}{dx} - y = 0$ is . . . . . . .

Vedclass Test Series

Exam Paper Generator

Online Exam Module