In my textbook, the authors said that a differential equation is linear if it can be expressed in the form

$$a_0(t)y^{(n)}+a_1(t)y^{(n-1)}+\cdots+a_n(t)y=g(t)$$

According to the definition, why the differential equation

$$y'''+2e^ty''+yy'=t^4$$

is nonlinear? $y$ is a function of t, $y'$ is also a function of t. We can view the term $yy'$ in the following way: $y$, a function of $t$ times the derivative $y'$. Therefore the diffrerential equation is linear. Am I right?

The point is that the $a_i$ and $g$ are fixed, known functions of $t$: the unknown function $y$ only appears in a linear combination of it and its derivatives.

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