Let's address each of the questions from the image.

Question 1:

The first equation given is of the form: $A u_{xx} + 2B u_{xy} + C u_{yy} = F(x, y, u, u_x, u_y)$ where $A$, $B$, and $C$ are functions of $x$ and $y$. It asks when the equation is said to be elliptic.

An equation of this form is classified based on the discriminant $\Delta = B^2 - AC$:

• Elliptic: If $AC - B^2 > 0$.
• Parabolic: If $AC - B^2 = 0$.
• Hyperbolic: If $AC - B^2 < 0$.

So, the correct condition for the equation to be elliptic is option (d) $AC - B^2 > 0$.

Question 2:

The second question asks about the type of the one-dimensional wave equation.

The one-dimensional wave equation is typically written as: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ This is a second-order linear partial differential equation, which is classified as hyperbolic.

So, the correct answer for this question is option (a) hyperbolic.

Do you have any questions or would you like further explanation on any of these topics? Here are 5 related questions for further exploration:

1. What is the geometric interpretation of an elliptic equation?
2. How can you differentiate between hyperbolic, parabolic, and elliptic PDEs geometrically?
3. What are some real-world examples of elliptic PDEs?
4. Can the classification of a PDE change depending on boundary conditions?
5. What are the differences in boundary conditions for hyperbolic, elliptic, and parabolic equations?

Tip:

Understanding the discriminant $B^2 - AC$ helps in classifying not just second-order PDEs, but also determining the nature of their solutions (e.g., propagating waves, diffusion, etc.).