# PDE1

Introduce the change of coordinates

So is now a function of . Recompute the derivatives with respect to these new coordinates.

,

.

Plug these derivatives into the original equation.

Choose any values of to make the problem easy to solve, with the caveat that . For example, let so that the term goes away. Then let so that the coefficient equals unity. Then,

Use the integrating factor

Here is an undetermined function of .

Now we determine the unique solution that corresponds to the inital condition . Using the general solution above, set .

But in the general solution, we have instead of . So making the appropriate substitutions,

Plug this function into the general solution.

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