Partial differential equations-heat equations

Question

Assignment:

Q1) Let A(x,y) be the area of a rectangle not degenerated of dimensions x and y, in a way that the rectangle is inside a circle of a radius of 10. Determine the domain and the range of this function.

Q2) The wave equation (c²∂² u / ∂ x² = ∂² u / ∂ t²) and the heat equation (c∂² u / ∂ x² = ∂ u / ∂ t) are two of the most important equations of physics (c is a constant). They are called partial differential equations. Show the following:

a) u = cos x cos ct and u = e^x cosh ct satisfies the wave equation.

b) u = e^-ct sin x and u = t^-1/2 e^[(-x^2)/(4ct)] satisfies the heat equation.

Provide complete and step by step solution for the question and show calculations and use formulas.