Why is Newton's second law with potentials not a linear equations?

Question

I was trying to learn Quantum physics by myself using MIT's 8.04 course. I came accross this equation:

I don't understand why the above is true. I understand the definition of linearity. But I don't know why two solutions to the above wouldn't be a solution if both sides involve derivatives (I am fully aware that one is a derivative wrt x and the other to t, but I don't know why that matters).
I don't understand why the above is not a linear equation and why adding/multiplying solutions doesn't work. Can someone give me an example?

J. Murray · Answer

Solutions to the equation $x'(t) = x^2(t)$
take the form
$$x(t) = frac{-1}{t+C}$$
for some integration constant $C$.  If I multiply $x(t)$ by some constant, does it remain a solution?  What if I add two valid solutions together?
The problem is the nonlinearity on the right-hand side of the differential equation.  If there were a linear function $V'(x(t))$ instead of the obviously-nonlinear quadratic function, then you can show quite easily that solutions can be  added and multiplied by constants to yield other solutions.  If the function is nonlinear, this is no longer true, as my simple example demonstrates.

shubham · Answer

It's just that the solution of the differential equations are not in single powers of $t$. So, the equation is a non liner equation since the solution to the differential equation is non linear.

Why is Newton's second law with potentials not a linear equations?

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