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Partial differential equation

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PDE Utt = Uxx+sin(3&#61552;x) 0<x<1 0<t<&#61605;

BC's U(0,t)=0
U(1,t)= 0

IC's U(x,0)= 2sin(2&#61552;x)
Ut(x,0)= 0

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https://brainmass.com/math/partial-differential-equations/partial-differential-equation-127759

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PDE Utt = Uxx+sin(3x) 0<x<1 0<t<

BC's U(0,t)=0
U(1,t)= 0

IC's U(x,0)= 2sin(2x)
Ut(x,0)= 0

We first must consider the homogenous equation:

We start as always with separating the wave equation:

Plugging it back into the homogenous equation we obtain:

Each side is independent of the other (since each side is a function of a single independent variable).

This can occur if an only if both sides equal the same constant number.

Hence:

The spatial equation becomes:

Case 1:

The general solution of the equation is:

Using boundary conditions we have:

Hence for we get only the trivial solution.

Case 2:

The general solution of the equation is:

Using boundary conditions we have:

Hence for we get only the trivial solution.

So we are left with case 3:

The general solution of the equation is:

...

Solution Summary

This provides an example of solving a partial differential equation. The functions of partial differential equations are examined.

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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 &#8706;^2 u / &#8706; x^2 = &#8706;^2 u / &#8706; t^2) and the heat equation (c &#8706;^2 u / &#8706; x^2 = &#8706; u / &#8706; 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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