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Consider the differential equation dy/dx = y³ / (2(x y² - x²)). Statement 1: The substitution z = y² converts this into a first-order homogeneous differential equation. Statement 2: Its solution is y² - x² = C. Choose the correct option.

1. Statement 1 is false and Statement 2 is true.
2. Both statements are true.
3. Statement 1 is true and Statement 2 is false.
4. Both statements are false.

Correct answer: Statement 1 is true and Statement 2 is false.

Solution

Putting z = y² indeed produces a first-order homogeneous equation in (x, z), so Statement 1 is true; however solving it does not give y² - x² = C, so Statement 2 is false.

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