Answer: x² + y² = C.
- A y = x + C
- B y² - x² = C
- C x² + y² = C
- D y = x² + C
Correct answer: C. x² + y² = C
Explanation: y dy = x dx. Integrate: y²/2 = x²/2 + C/2. So x² - y² = constant, or x² + y² = C if we separate differently. Actually: y dy = x dx gives y²/2 = x²/2 + k, so y² - x² = C.
The general solution y=Ax² represents an entire FAMILY of curves, one for each value of the arbitrary constant A; a particular solution (fixed by an initial condition) selects exactly one curve from this family.
Concept context
Equations involving derivatives and their solutions