Consider the following statements:I. y = xe^ 2x is the solution of...

For Statement I:

Given $y = xe^{2x}$

Differentiating with respect to $x$:

$\dfrac{dy}{dx} = e^{2x} + 2xe^{2x} = e^{2x}(1 + 2x)$

Also, $y\left(2 + \dfrac{1}{x}\right) = xe^{2x} \left(\dfrac{2x + 1}{x}\right) =$e^{2x}(2x + 1)$<br><br>Since \dfrac{dy}{dx} =$ y\left(2 + \dfrac{1}{x}\right), Statement I is correct.

For Statement II:

Given $y = x\ln|x| + cx$

Differentiating with respect to $x$:

\dfrac{dy}{dx} = \ln|$x| + x\left(\dfrac{1}{x}\right) +$ c = \ln|x| + 1 + c$<br><br>Also, \dfrac{x + y}{x} =1 + \dfrac{y}{x} =1 + \dfrac{x\ln|x| + cx}{x} =1 + \ln|x| + c$

Since $\dfrac{dy}{dx} = \dfrac{x + y}{x},Statement II is correct.<br><br>Both statements I$ and II are correct.

Answer: Both I and II