Which of the following is a homogeneous differential equation?
Explanation
A differential equation of the form dy/dx = F(x, y) is said to be homogeneous if F(x, y) is a homogeneous function of degree zero. This means when we replace x by λx and y by λy, we should get F(λx, λy) = λ⁰F(x, y) = F(x, y). Let me analyze each option:
Option (A): The equation has constant terms (5 and 4), so it's NOT homogeneous.
Option (B): Rearranging gives dy/dx = xy/(x³ + y³). Here numerator has degree 2, denominator has degree 3, so overall degree is -1 0. NOT homogeneous.
Option (C): Rearranging gives dy/dx = -(x³ + 2y²)/(2xy). The terms in numerator have different degrees (x³ has degree 3, y² has degree 2). NOT homogeneous.
Option (D): Rearranging gives dy/dx = y²/(x² - xy - y²). All terms have degree 2. F(λx, λy) = λ²y²/[λ²(x² - xy - y²)] = λ⁰F(x, y). This IS a homogeneous differential equation.
Solution Steps
Step 1: Recall that a homogeneous differential equation has F(x,y) as a homogeneous function of degree zero.
Step 2: For each option, express in the form dy/dx = F(x,y) and check if F(λx, λy) = λ⁰F(x,y).
Step 3: Option (D): y² dx + (x² - xy - y²) dy = 0 gives dy/dx = y²/(x² - xy - y²).
Step 4: Verify: F(λx, λy) = λ²y²/[λ²(x² - xy - y²)] = λ⁰F(x, y). Hence homogeneous.
Step 5: Options (A), (B), (C) do not satisfy the homogeneous condition.