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# Solving Partial Differential Equations

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1. Solve

u(0,t) = u(,t) =0

u(x,0) = xsinx ,

https://brainmass.com/math/partial-differential-equations/solving-partial-differential-equations-117456

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1. Solve

u(0,t) = u(,t) =0

u(x,0) = xsinx ,

We start by setting the function u(x,t) as a product of a single-variable functions:

Therefore the partial derivatives become:

Plugging it back into the equation, one obtains:

Dividing by separates the equation:

Since both sides are totally independent (each is a function of an ...

#### Solution Summary

A PDE is 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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## Solving Partial Differential Equations by Change of Variables

In solving this problem, derive the general solution of the given equation by using an appropriate change of variables.

1. ∂u/∂t - 2 ∂u/∂x = 2

Answer: u(x,t) = f(x + 2t) - x

In this exercise, (a) solve the given equation by the method of characteristic curves, and (b) check you answer by plugging it back into the equation.

2. ∂u/∂x + x2 ∂u/∂y = 0

Answer: (a) u(x,y) = f(1/3 x3 - y)

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