Consider the following differential equations:
(A) \((x^2 + 3y^2)\,dx - xy\,dy = 0\)
(B) \((x + y + 1)\,dy = (x - y)\,dx\)
(C) \(\displaystyle\frac{dy}{dx} = \frac{x^2y}{x^3 + y^3}\)
(D) \(\displaystyle\frac{dy}{dx} = \frac{x^2 + y}{xy}\)
Which of the above are homogeneous differential equations?
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
Writing each equation as \(\frac{dy}{dx} = F(x, y)\), it is homogeneous when F is a homogeneous function of degree zero, that is, when every term of the numerator and of the denominator has the same degree.
\( \displaystyle \text{(A)}\quad F = \frac{x^2 + 3y^2}{xy} \)
\( \displaystyle \text{(B)}\quad F = \frac{x - y}{x + y + 1} \)
\( \displaystyle \text{(C)}\quad F = \frac{x^2y}{x^3 + y^3} \)
\( \displaystyle \text{(D)}\quad F = \frac{x^2 + y}{xy} \)
So F has degree zero in (A) and (C), while (B) has a constant term and the numerator of (D) mixes degrees 2 and 1.
Checking the statements
(A) \((x^2 + 3y^2)\,dx - xy\,dy = 0\) — Correct. Numerator and denominator both of degree 2.
(B) \((x + y + 1)\,dy = (x - y)\,dx\) — Incorrect. The constant 1 has degree 0, unlike x and y.
(C) \(\frac{dy}{dx} = \frac{x^2y}{x^3 + y^3}\) — Correct. Every term has degree 3.
(D) \(\frac{dy}{dx} = \frac{x^2 + y}{xy}\) — Incorrect. \(x^2\) has degree 2 but y has degree 1.