Question

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

• A. Ce x2/2
• B. Cex2
• C. Ce2x2
• D. Cex2/3

Answer 1

The Correct Option is A

We are given the differential equation: \[ x \frac{dy}{dx} + y = x \frac{f(xy)}{f'(xy)} \] Step 1: Make a substitution Let \( u = xy \). Then, by the chain rule, we have: \[ \frac{du}{dx} = y + x \frac{dy}{dx} \] This transforms the given equation into: \[ x \frac{dy}{dx} + y = x \frac{f(u)}{f'(u)} \] Using the substitution \( u = xy \), this simplifies to: \[ \frac{du}{dx} = \frac{f(u)}{f'(u)} \] Step 2: Solve the equation Now we separate the variables and integrate: \[ \frac{f'(u)}{f(u)} du = \frac{1}{x} dx \] Integrating both sides: \[ \ln |f(u)| = \ln |x| + C \] Thus, we have: \[ |f(u)| = Cx \] Since \( u = xy \), we can substitute back to get: \[ |f(xy)| = Ce^{x^2/2} \] Conclusion Therefore, the answer is: \[ \boxed{Ce^{x^2/2}} \]