Consider the following statements: I. y=xe^{2x} is the solution of \frac{dy}{dx}=y\left(2+\frac{1}{x}\right). II. y=x\ln|x|+cx is the solution of \frac{dy}{dx}=\frac{x+y}{x}. Which of the statements given above is/are correct?

Explanation

For Statement I: differentiating y=xe^{2x} yields \frac{dy}{dx} = e^{2x} + 2xe^{2x} = e^{2x}(1 + 2x) = \frac{xe^{2x}}{x}(1+2x) = y\left(\frac{1}{x} + 2\right), so it is correct. For Statement II: \frac{dy}{dx} = 1 + \frac{y}{x}. Let y=vx, then v + x\frac{dv}{dx} = 1 + v \implies dx/x = dv. Integrating gives \ln|x| + c = v = \frac{y}{x} \implies y = x\ln|x| + cx, so Statement II is also correct.

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