Here are the answers for the questions in the image:

1. The one-dimensional heat equation is:

   • The one-dimensional heat equation is classified as a parabolic partial differential equation. Therefore, the correct answer is (b) parabolic.

2. The two-dimensional Laplace equation is:

   • The Laplace equation is classified as an elliptic partial differential equation. Thus, the correct answer is (d) elliptic.

3. The equation $4u_{xx} - u_{yy} = 0$ is:

   • To classify this equation, we compare it to the general form $Au_{xx} + Bu_{xy} + Cu_{yy} = 0$. Here, $A = 4$, $B = 0$, and $C = -1$.
   • The discriminant is given by $B^2 - 4AC = 0^2 - 4(4)(-1) = 16$, which is positive, indicating a hyperbolic equation. Hence, the answer is (a) hyperbolic.

Let me know if you need further explanations, or if you have any questions!

Here are five related questions for deeper exploration:

1. What is the general form of a heat equation in two dimensions?
2. How can we classify a partial differential equation based on its discriminant?
3. What are some physical phenomena modeled by elliptic equations?
4. What is the significance of boundary conditions in solving parabolic equations?
5. How do hyperbolic equations differ in their solutions compared to parabolic and elliptic equations?

Tip: To classify a second-order partial differential equation, compare it to the standard form $Au_{xx} + Bu_{xy} + Cu_{yy} = 0$ and evaluate the discriminant $B^2 - 4AC$.