If , and . Then, is equal to
- A.3
- B.2
- C.1
- D.0
Show answer and solution
Correct option: A
Solution
Rewrite as x dy - y dx = y² dy. Divide both sides by y²: (x dy - y dx)/y² = dy, which gives d(x/y) = dy. Integrating: x/y = y + C. Apply y(1) = 1: 1/1 = 1 + C, so C = 0. Thus x/y = y, giving y² = x. At x = -3: y² = -3 is impossible unless we recheck — actually x = y², so at x = -3, y² = -3. Wait, let me redo: x/y = y means x = y². At x = -3, this gives no real solution, but the problem states y(-3) = 3, so let me recheck the sign.
Actually from d(x/y) = dy, integrating gives x/y = y + C. With y(1)=1: C = 0, so x = y². Then y(-3): y² = -3 has no real solution. Let me try the other direction: (x dy - y dx)/y² = dy means -d(x/y) isn't right — actually d(x/y) = (y dx - x dy)/y², so x dy - y dx = -y² d(x/y). So the equation gives -y² d(x/y) = y² dy, meaning d(x/y) = -dy. Integrating: x/y = -y + C. With y(1)=1: 1 = -1 + C, so C = 2. Thus x/y = 2 - y, giving x = 2y - y² = -(y-1)² + 1
Actually from d(x/y) = dy, integrating gives x/y = y + C. With y(1)=1: C = 0, so x = y². Then y(-3): y² = -3 has no real solution. Let me try the other direction: (x dy - y dx)/y² = dy means -d(x/y) isn't right — actually d(x/y) = (y dx - x dy)/y², so x dy - y dx = -y² d(x/y). So the equation gives -y² d(x/y) = y² dy, meaning d(x/y) = -dy. Integrating: x/y = -y + C. With y(1)=1: 1 = -1 + C, so C = 2. Thus x/y = 2 - y, giving x = 2y - y² = -(y-1)² + 1
[AIEEE 2005]