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

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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

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$Ce^{x^2/2}$

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

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