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

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?

• Forget about quantum mechanics, and consider a spring and mass system where the spring is nonlinear - for example the spring force is $-Kx - ax^3$. (That example is well known - google for the Duffing equation). – alephzero Aug 14 '20 at 2:50
• Welcome to Phys.SE. Avoid pasting images of text, type in the necessary content for your question in order to make the contents searchable.Use LaTeX/MathJax to typeset equations. – ohneVal Jan 21 at 8:38

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.
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.
• No, non-linear here means non-linear in $x(t)$. There are plenty of linear differential equations whose solutions are not linear, like $\dot x(t)=x(t)$. – kaylimekay Jan 21 at 8:20