Question

a)Determine the order of the following PDE, and state whether it is linear or nonlinear, homogeneous or nonhomogeneous: (uₓₓ+uy)ₜₜ+e²ʸ+xyuₓy=0 b) Determine whether the following second order PDE is parabolic, hyperbolic or elliptic: (uₓₓ+2yy+3uₓy+2uₓ+3uy=4x³u c)Use the method of separation of variables to solve the following PDE: uₓ=e-ᵗcosx with u(x,0)=0 and ∂u(0,t)/∂t=0

Answer 1

a) The given PDE is of second order and it is nonlinear and nonhomogeneous.

b) The second order PDE provided is elliptic.

Step-by-step explanation:

a) The equation provided is of second order due to the highest derivative being a second derivative in both space and time. It is nonlinear as it contains terms that involve products of the unknown function u and its derivatives. Additionally, it's nonhomogeneous due to the presence of the term
`\(e^(2y)\)`.

b) To determine the nature of the second order PDE, it's essential to examine its characteristics. The presence of second-order derivatives both in x and y suggests an elliptic equation. In this case, the equation is elliptic due to the nature of its terms.

In part c, solving the PDE
`u_x = e^(-t)`cos(x) with the initial conditions u(x, 0) = 0 and
`(\partial u(0, t))/(\partial t)` = 0 involves separation of variables. This method involves expressing the solution as a product of functions, one depending only on x and the other only on t. By isolating variables and integrating, the solution will take form.

First, expressing u(x, t) as X(x)T(t), then separating variables and solving for each part independently yields X(x) = Asin(x) + Bcos(x)\) and T(t) = C
`e^(-t)\)` after applying the initial conditions. Finally, combining both parts with appropriate constants yields the solution u(x, t) = B
`e^(-t)`cos(x), fulfilling the given PDE and initial conditions.

This method allows the solution to be deduced step by step, ensuring each component satisfies the PDE and the provided initial conditions without ambiguity.