Solve ut + xux = 0 with the initial condition u(x, 0) = e−x2 .

1.	Solve ut + xux = 0 with the initial condition u(x, 0) = e−x2 .
Answer» The equation for the characteristic curves are dt/1 = dx/x . Thus the characteristic curves are given by t = ln x + c', or xe^−t = c. Therefore, the general solution of the equation is u(x, t) = f(xe^−t). To determine f, we set t = 0 and find that f(x) = e^−x2. Thus u(x, t) = e−x²e^−2t

Answer»

The equation for the characteristic curves are dt/1 = dx/x .

Thus the characteristic curves are given by

t = ln x + c', or

xe^−t = c.

Therefore, the general solution of the equation is

u(x, t) = f(xe^−t).

To determine f, we set t = 0 and find that

f(x) = e^−x2.

Thus u(x, t) = e−x²e^−2t

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Solve ut + xux = 0 with the initial condition u(x, 0) = e−x2 .

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