Consider the differential equation \(\displaystyle\frac{dy}{dx} = -\frac{x}{y}\), \(y \gt 0\), and the following statements.
(A) Its order is 1 and its degree is 1
(B) Its general solution is \(x^2 - y^2 = C\), C an arbitrary constant
(C) Its solution curve through the point (3, 4) also passes through the point (0, 5)
(D) Its solution curve through the point (1, 2) also passes through the point (2, 2)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
The equation is variables separable:
\(y\,dy = -x\,dx\)
\( \displaystyle \frac{y^2}{2} = -\frac{x^2}{2} + C_1 \)
\(x^2 + y^2 = C\)
The curve through (3, 4) has \(C = 9 + 16 = 25\); the curve through (1, 2) has \(C = 1 + 4 = 5\).
So the order and degree are 1 and 1, the general solution is \(x^2 + y^2 = C\), and the two solution curves are \(x^2 + y^2 = 25\) and \(x^2 + y^2 = 5\).
Checking the statements
(A) order 1 and degree 1 — Correct. Only \(\frac{dy}{dx}\) appears, with power 1.
(B) general solution \(x^2 - y^2 = C\) — Incorrect. The general solution is \(x^2 + y^2 = C\).
(C) curve through (3, 4) passes through (0, 5) — Correct. \(0 + 25 = 25\), and \(y = 5 \gt 0\).
(D) curve through (1, 2) passes through (2, 2) — Incorrect. \(4 + 4 = 8 \ne 5\).