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Sticky Note
Which of the following equations is of the type 𝑦 = 𝑥 𝑑 𝑦 / 𝑑 :: 𝑥 + 𝑓 𝑑 𝑦 /𝑑 𝑥 ​?
  1. d 2 y d x 2 ​ and d y d x ,
  2. dy /dx​ =x+f
  3. y=xdy /dx​ +f
  4. dy /dx​ =x dy /dx​+f
Explanation

The equation of the type y=xdydx+fdydxy = x frac{dy}{dx} + f frac{dy}{dx}  can be identified by comparing it to the differential equation form:

y=xd2ydx2+fdydxy = x frac{d^2y}{dx^2} + f frac{dy}{dx}

In this form, the equation involves both the second derivative d2ydx2frac{d^2y}{dx^2} and the first derivative dydxfrac{dy}{dx} with coefficients xx and ff respectively.

Therefore, among the options provided, the equation that fits this type is:

y=xd2ydx2+fdydxy = x frac{d^2y}{dx^2} + f frac{dy}{dx}

This equation indicates a relationship where yy is expressed as a combination involving both d2ydx2frac{d^2y}{dx^2} and dydxfrac{dy}{dx}, with coefficients xx and ff respectively.

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