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*Partial Differential Equations, 2nd edition*in an attempt to prepare for my upcoming class on Partial Differential Equations but this problem has me stumped. I feel like it should be fairly simple, but I just can't get it.

*10. Solve ##u_{x} + u_{y} + u = e^{x+2y}## with ##u(x,0) = 0##.*(

*Partial Differential Equations, 2nd ed.*by Walter A. Strauss, pg. 10)

This section features only a few exercises and I assume they are to be solved using the methods presented in this section (primarily because they appear so early in the text). This section introduces two types of equations, the constant coefficient equation (which the above question is an example of) and the variable coefficient equation.

To solve the constant coefficient equation Strauss introduces the Coordinate Method. In general, the equation ##au_{x} + bu_{y} = 0## can be simplified by using the substitutions ##x' = ax + by## and ##y' = bx - ay## (note: these are

*not*the derivatives of x and y, but are simply new variables - I'm sticking with the notation used in the text). By using this change of variables the equation ##au_{x} + bu_{y}## is reduced to ##(a^{2} + b^{2})u_{x'}##.

If I apply this same method to the above exercise, I find:

##u_{x} + u_{y} + u = e^{x+2y}##

##2u_{x'} + u = ?##

**My difficulty lies in writing ##e^{x+2y}## in terms of only ##x'## and ##y'##.**I feel like I'm missing something fairly obvious but can't just give up on it.

Otherwise, I have reduced this to an ODE solvable by using the method of integrating factors.

Any help would be greatly appreciated! Thanks!