I think I found a small error in Landau & Lifschitz "Mechanics" (3rd ed.). In section 28 (Anharmonic oscillations), they are discussing how to solve the following anharmonic oscillator problem:
$$\ddot{x}+\omega_0^2 x = -\alpha x^2-\beta x^3 \tag{28.9}$$
They show how one can solve the anharmonic oscillator problem perturbatively by expanding the solution in powers of the initial amplitude $a$ and supposing that the fundamental frequency is also shifted, the shift being given by another power series in the amplitude. The ansatz for the lowest-order solution is the following:
$$x^{(1)}=a\cos \omega t \tag{28.10}$$
One can grind out the higher order equations and arrive at the equation for the 3rd order perturbation:
$$\ddot{x}^{(3)}+\omega_0^2 x^{(3)}=-2\alpha x^{(1)}x^{(2)}-\beta x^{(1)3}+2\omega_0 \omega ^{(2)} x^{(1)}\tag{pg. 87}$$
Now, first I want to mention that in that equation we have $\beta x^{(1)3}$ which we can write out as
$$\begin{align}\beta x^{(1)3}&=\beta a^3\cos^3\omega t\\ &=a^3\left(\color{red}{\frac{1}{4}\beta} \cos 3\omega t \color{red}{+ \frac{3}{4}\beta} \cos \omega t\right) \end{align}$$
But in the next line the following full expansion is given (where I highlight the error in red):
$$\begin{align} \ddot{x}^{(3)}+\omega_0^2 x^{(3)}=a^3&\left[\color{red}{\frac{1}{4}\beta}-\frac{\alpha^2}{6\omega_0^2}\right]\cos 3\omega t +\\ &+a\left[2\omega_0\omega^{(2)}+\frac{5a^2\alpha^2}{6\omega_0^2}\color{red}{- \frac{3}{4}a^2\beta}\right]\cos\omega t \tag{pg. 87}\end{align}$$
So the it looks like a spurious minus sign has entered, which carries on throughout the rest of the section.
[Question]: Is this actually a mistake? I can't find any errata for the book so I can't validate this.