Linear vs nonlinear diff equation

[Calpalned]

How can one tell if an equation is linear or not? Is it necessary to memorize what the graph looks like?
View attachment 90518

 

[Geofleur]

The equation is linear if there are no terms having products of $y$ with itself or with any of its derivatives. Also, there must no terms having products of derivatives of $y$ with other derivatives of $y$. So, for example, the presence of $y^2$ or of $y\frac{dy}{dx}$ would signal nonlinearity, as would the presence of $\frac{d^2 y}{dx^2}\frac{dy}{dx}$.

The ultimate test for linearity is this: If $y_1$ and $y_2$ are both solutions of the differential equation, then so must be $a y_1 + b y_2$, where $a$ and $b$ are constants.

 

[Calpalned]

The equation is linear if there are no terms having products of $y$ with itself or with any of its derivatives. Also, there must no terms having products of derivatives of $y$ with other derivatives of $y$. So, for example, the presence of $y^2$ or of $y\frac{dy}{dx}$ would signal nonlinearity, as would the presence of $\frac{d^2 y}{dx^2}\frac{dy}{dx}$.

The ultimate test for linearity is this: If $y_1$ and $y_2$ are both solutions of the differential equation, then so must be $a y_1 + b y_2$, where $a$ and $b$ are constants.

thank you