Why is this first-order ODE considered non-linear?

Question

Consider the following equation: $$y'=\sin y.$$

What causes this equation to be considered non-linear? $y'$ is not multiplied by a power, and neither is $\sin y$.

Answer 1

A linear differential equation is one that has the following form: $$ \frac{dy}{dx}+p(x)y=q(x).$$ To see that this doesn't hold in this case, try to convert $$ \frac{dy}{dx}=\sin y$$ to the appropriate form. Rearranging we have $$ \frac{dy}{dx}-(1)\sin y=0.$$ As you can see, this is not linear in $y$. That is, $y$ does not sit on its own with a function of $x$ as its prefactor in the non-differential term.

Answer 2

A first-order differential equation is considered linear if it can be written in the form

$$y'+p(x)y=f(x)$$

$y'=\sin y$ can't be written in that format

Answer 3

Because:

$y^\prime = y-\frac{y^3}{6}+\frac{y^5}{120}-\frac{y^7}{5040}+O\left(y^9\right)$.

Answer 4

"Nonlinear" is NOT just a matter of powers or polynomials in y. sin(y) is itself a "non-linear" function of y.