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

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Correct answer: C. Both I and II

For Statement I: Given y = xe^ 2x Differentiating with respect to x: dy dx = e^ 2x + 2xe^ 2x = e^ 2x (1 + 2x) Also, y (2 + 1 x ) = xe^ 2x ( 2x + 1 x ) = e^ 2x (2x + 1) Since dy dx = y (2 + 1 x ), Statement I is correct. For Statement II: Given y = x |x| + cx Differentiating with respect to x: dy dx = |x| + x ( 1 x ) + c = |x| + 1 + c Also, x + y x = 1 + y x = 1 + x |x| + cx x = 1 + |x| + c Since dy dx = x + y x , Statement II is correct. Both statements I and II are correct. Answer: Both I and II

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