关于判断指定常微分方程（ODEs）为Homogeneous或Nonhomogeneous的技术咨询

Hey there! Let's walk through each of these ODEs step by step to classify them as homogeneous or nonhomogeneous. First, a quick recap of the key definitions to keep in mind:

• For linear ODEs: An equation is homogeneous if every term contains the dependent variable (here, y) or one of its derivatives. There are no standalone terms (like constants or functions of only the independent variable x) that don't involve y.
• For nonlinear ODEs: The rule is a bit stricter—we check if replacing y with ky (where k is any constant) lets us factor out a uniform power of k from the entire equation. If we can't do this, the ODE is nonhomogeneous.

Now let's apply this to each equation:

1. Equation: y'' - y' = y
   First, rearrange it to standard form: y'' - y' - y = 0
   This is a linear ODE, and every term involves y or its derivatives. There are no terms that don't depend on y, so this is Homogeneous.

2. Equation: y'' - y' = sin(y)
   Rearranged: y'' - y' - sin(y) = 0
   This is a nonlinear ODE (thanks to the sin(y) term, which isn't a linear function of y). Let's test the homogeneous check: replace y with ky. The left-hand side becomes (ky)'' - (ky)' - sin(ky) = k y'' - k y' - sin(ky). Since sin(ky) ≠ k sin(y) for any constant k (except trivial edge cases), we can't factor out a uniform k from the entire expression. So this is Nonhomogeneous.

3. Equation: y'' - y' = xy
   Rearranged: y'' - y' - xy = 0
   This is a linear ODE. Even though there's an x (the independent variable) multiplied by y, the term xy still depends on the dependent variable y. All terms in the equation involve y or its derivatives, so this is Homogeneous.

备注：内容来源于stack exchange，提问作者JMT