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Partial differential equations

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A) Classify and find general expressions for the characteristic coordinates for the equation {see attachment}
b) Use the canonical coordinates {see attachment} and transfer the above PDE into the new coordinates. Solve it in the new coordinates and show that {see attachments} where F and G are arbitrary functions of their arguments.

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a) Classify and find general expressions for the characteristic coordinates for the equation

( 1)

b) Use the canonical coordinates and and transfer the above PDE into the new coordinates. Solve it in the new coordinates and show that

where F and G are arbitrary functions of their arguments.

Solution:
a) The quadratic terms of the equation (1) are:

The corresponding characteristic equation is:
( 2)
( 3)
Since , the PDE is of hyperbolic type and ...

Solution Summary

This shows how to classify and find general expressions for characteristic coordinates and use given canonical coordinates to transfer a partial differential equation to new coordinates.

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Partial Differential Equations : Heat Equations

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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