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A first-order equation is homogeneous when it can be written as \( dy/dx = \)

  • A.a function of \( y/x \) alone
  • B.a function of \( x \) alone
  • C.a function of \( y \) alone
  • D.a product of a function of \( x \) and one of \( y \)

Answer: A

If the right side depends only on the ratio, the substitution \( y = vx \) makes the equation separable in \( v \) and \( x \). A function of \( x \) alone integrates directly; the product form is the separable case; and a function of \( y \) alone is separable too. Homogeneity is specifically about the ratio

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