The image you provided contains various statements related to differential equations, partial differential equations, and their respective orders and degrees. Here's a breakdown of the points mentioned:

1. Order and Degree of a Differential Equation: The first equation provided is: $\left( \frac{d^2 y}{dx^2} \right)^3 + 5 \left( \frac{dy}{dx} \right)^3 + 6y = 0$

   • This equation involves the second derivative $\frac{d^2y}{dx^2}$ and the first derivative $\frac{dy}{dx}$, so its order is 2 (since the highest derivative is $\frac{d^2y}{dx^2}$).
   • The degree of the equation is 3, because the highest power of the highest derivative is cubed in the term $\left(\frac{d^2y}{dx^2}\right)^3$.

2. Solution of a Differential Equation: $y = \tan(x + c)$ is a solution to the differential equation: $\frac{dy}{dx} = 1 + y^2$

   • Here, $y = \tan(x + c)$ is indeed a solution to the given differential equation, as it satisfies the relation when differentiated.

3. Eliminating Constants from a Partial Differential Equation: The relation given is: $z = ax + by + a^2 + b^2$ By eliminating the arbitrary constants $a$ and $b$, you obtain the partial differential equation: $4z = p^2 + q^2$ where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$. The equation represents a first-order partial differential equation.

4. Second Derivative Calculation: The equation is: $w = x + 2y + z^2$ The second derivative with respect to $x$ is: $\frac{\partial^2 w}{\partial x^2} = \frac{\partial w}{\partial x \partial y}$ This relationship is provided for a specific form of a function.

If you'd like further clarification or step-by-step explanations of any particular section, feel free to ask! Here are some related questions you might find interesting:

1. How do we calculate the order and degree of higher-order differential equations?
2. What is the method for solving non-linear differential equations like $\frac{dy}{dx} = 1 + y^2$?
3. Can you explain the process of eliminating arbitrary constants in partial differential equations with an example?
4. What is the significance of the second derivative in partial differential equations?
5. How do we handle cases where the second derivative of a function with respect to two variables is involved?

Tip: When calculating the degree of a differential equation, always make sure to express all terms in polynomial form and avoid fractional powers or non-polynomial expressions for the derivatives.