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Partial Differential Equations : Heat Equations

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1) 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.

2) The wave equation (c^2 ∂^2 u / ∂ x^2 = ∂^2 u / ∂ t^2) and the heat equation (c ∂^2 u / ∂ x^2 = ∂ 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.

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Partial differential equations problems
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I need the solution for this problems and the steps of how it was done, even if its handwritten.

1) 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.

We can put the center of the rectangle at the same center of the circle. See the graph above. We can set up an inequality as follows. Assume the dimensions of this rectangle are x and y. Hence,
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PDE problems are solved. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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