PDE by SL method

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Math/Physics

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PDE's

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Partial Differential Equations Problems

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.

Partial differential equation problem

According to the ideal gas law, the pressure, the temperature and the volume of a gas are related by PV = kT (k is a constant). Show the following: (attached)