About definition of linear differential equation

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Calculus

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I have some questions regarding the definition of a linear ordinary differential equation. By definition, a linear ODE has the following form:

https://preview.redd.it/wtqay4b99fcd1.png?width=555&format=png&auto=webp&s=57d45fd40ea6e04191682060690186a102460463

where aₙ(x) is not necessarily linear.

Based on this definition, can aₙ(x)=y(x)? If it can be, then my question is resolved. It's in the other case, where it cannot occur, where my question lies. Suppose I have an equation of the following form:

https://preview.redd.it/9idnjdqkcfcd1.png?width=614&format=png&auto=webp&s=41d0518ca8cf3aaee03ee1bef5ac1e311211bf34

Now suppose we find the family of solutions y(x) = x² k. I'm not sure if a solution like this makes sense, but let's assume it does.

Now with these assumptions:

https://preview.redd.it/0t07iogpdfcd1.png?width=621&format=png&auto=webp&s=2efe7f6f146a2756b23689eae6aacc898941ea9c

Then, this form would no longer be a linear equation, so was it never linear?

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