# Linear vs nonlinear diff equation

How can one tell if an equation is linear or not? Is it necessary to memorize what the graph looks like?
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Geofleur
Gold Member
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.

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