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Examples of solutions to linear differential equation are (from Wikipedia): exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. So in this sense these are linear functions (for some generalized notion of "linear").

But I wonder if there are examples of functions that are "strictly nonlinear", in the sense that they are the solution of some nonlinear differential equation but not of any linear differential equation.

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  • $\begingroup$ I think you want to restrict the coefficients of the linear DE perhaps to polynomials in $x$. $\endgroup$
    – Somos
    Jun 1 at 18:29

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Those functions are not linear in any sense I can think of. Any of the functions you mentioned, while they are solutions of linear differential equations, they are equally well solutions of nonlinear equations. Likewise, if you produce a solution to a nonlinear equation, it could equally well also satisfy a linear equation.

For example, $y(x)=\sin(x)$ solves the nonlinear differential equation $y'(x)^2 = \cos^2(x)$ as well as linear equation $y''(x)=-y'(x)$. Any function $f(x)$ satisfies the linear differential equation $y'(x)=f'(x)$ and the nonlinear equation $y'(x)^2 = f'(x)^2$.

I'm really not sure your question is well-posed.

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