Here are the questions from your image for problems 2, 3, and 4:

2) If $y^{\frac{1}{m}} + y^{\frac{1}{n}} = 2x$, prove that: $(x^2 - 1) y_{n+2} + (2n+1) x y_{n+1} + (n^2 - m^2) y_n = 0$

3) If $\cos^{-1} \left( \frac{y}{b} \right) = \log \left( \frac{x}{u} \right)^n$, prove that: $x^2 y_{n+2} + 2 (2n+1) x y_{n+1} + 2n^2 y_n = 0$

4) If $y = \sin \left( \frac{\log y}{a} \right)$, then prove that: $(1 - x^2) y_{n+2} - (2n+1) x y_{n+1} - (n^2 + a^2) y_n = 0$ And find $(y_n)_0$.

Let me know if you'd like detailed solutions or explanations for any of these problems!

Here are some follow-up questions:

1. What methods can be used to solve these types of differential equations?
2. How does the relationship between the variables change the nature of the solutions?
3. Can you explain the role of boundary conditions in solving these differential equations?
4. How can the initial conditions help determine $(y_n)_0$ in question 4?
5. What significance do terms like $(n^2 - m^2)$ or $(n^2 + a^2)$ hold in these equations?

Tip: When proving identities or solving differential equations, simplifying the equation step by step using known identities and techniques (such as the chain rule or integration by parts) can be very helpful.