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I have this equation: $$\frac{d^2y}{dx^2}=\frac{dy}{dx}-2(3-y)y$$ which apparently has weak non-linearity and reduces to this: $$\frac{d^2y}{dx^2}=\frac{dy}{dx}-6y$$

It is not at all clear to me why this is the case. If you integrate the equation twice you will get a quadratic equation, right? So what linearity is the author speaking about?

Now this question might sound obtuse to you but keep in mind that I have not yet been exposed to a proper differential equations class. For the moment I just need to know how to reduce weakly non-linear differential equations.

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the original equation is $\frac{d^2y}{dx^2}=\frac{dy}{dx}=-6y+2y^2$ and with linearisation means the ausor that $\frac{dy}{dx}-6y+2y^2\approx \frac{dy}{dx}-6y$ i think, but we must know more about the situation. the second equation is easy to solve, the solution is $y \left( x \right) ={\it \_C1}\,{{\rm e}^{x/2}}\sin \left( 1/2\,\sqrt {23}x \right) +{\it \_C2}\,{{\rm e}^{x/2}}\cos \left( 1/2\,\sqrt {23}x \right) $ the first one is nonlinear and not so easy to solve.

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