Question

Which of the following is not a homogeneous differential equation?

• A. $\frac{dy}{dx}=\frac{y^2}{x^2+xy}$
• B. $\frac{dy}{dx}=\frac{y}{x}-\frac{y^3}{x^3}$
• C. $\frac{dy}{dx}=\frac{y^2}{x-1}$
• D. $\frac{dy}{dx}=\frac{2x-3y}{7x+4y}$

Answer 1

Answer: (C)

C

Answer 2

A differential equation $\frac{dy}{dx} = f(x,y)$ is homogeneous if $f(x,y)$ is a homogeneous function of degree zero. This means $f(tx, ty) = f(x,y)$ for any non-zero $t$. For rational functions, this implies the numerator and denominator must be homogeneous functions of the same degree.

• A: Numerator $y^2$ (degree 2), Denominator $x^2+xy$ (all terms degree 2). Homogeneous.
• B: Can be written as a function of $y/x$ (e.g., $y/x - (y/x)^3$). Homogeneous.
• C: Numerator $y^2$ (degree 2), Denominator $x-1$ (terms of degree 1 and 0). Denominator is not homogeneous. Thus, not homogeneous.
• D: Numerator $2x-3y$ (all terms degree 1), Denominator $7x+4y$ (all terms degree 1). Homogeneous.