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$$\frac{dy}{dx}+\sin x=5$$ $$\frac{dy}{dx}+\sin y=5$$

Questions:

Why is the first equation called linear differential equation and the second non-linear differential equation?

Please, I need a satisfying answer and thank you.

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In general, a differential equation is considered linear if:

  • the differential equation is of the form $$a_n(x)y^{(n)}(x)+a_{n-1}(x)y^{(n-1)}(x)+\dots+a_1(x)y'(x)+a_0(x)y(x)=g(x),\quad (1)$$
  • $y(x)$ is not multiplied by any of its derivative, $y'(x), y''(x),\dots, y^{(n)}(x)$
  • $y(x),y'(x),\dots, y^{(n)}(x)$ are all only to the first power.

All other equations that cannot be written in $(1),$ are non-linear differential equations.

Note

$a_0(x),\dots, a_n(x)$ and $g(x)$ can be zero or non-zero, constant or non-constant, or linear or non-linear. $y(x),y'(x),\dots, y^{(n)}(x)$ are the only functions that are used in determining if a differential equation is linear. I hope this answers your question.

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