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**Random Variable** I want to find the general solution to $\displaystyle \frac{\partial u}{\partial t} - 2 \ \frac{\partial u}{\partial x} = 2 $

so I let $\displaystyle \alpha = ax + bt $ and $\displaystyle \beta = cx+dt$

then $\displaystyle \frac{\partial u}{\partial t} = \frac{\partial u}{\partial \alpha} \frac{\partial \alpha}{\partial t} + \frac{\partial u}{\partial \beta} \frac{\partial \beta}{\partial t} = a \frac{\partial u}{\partial \alpha} + c \ \frac{\partial u}{\partial \beta} $

similarly, $\displaystyle \frac{\partial u}{\partial x} = b \frac{\partial u}{\partial \alpha} + d \ \frac{\partial u}{\partial \beta} $

so $\displaystyle (b-2a) \ \frac{\partial u}{\partial \alpha} + (d-2c) \ \frac{\partial u}{\partial \beta} =2 $

Would be OK to now do any of the following:

Let a=1,b=0,c=1, and d=2 (then $\displaystyle \frac{\partial u}{\partial \alpha} = -1 $)

Let a=0,b=1, c=1,and d=2 (then $\displaystyle \frac{\partial u}{\partial \alpha} = 2 $ )

Let a=1, b=0, c=1/2, d=1 (then then $\displaystyle \frac{\partial u}{\partial \alpha} = -1 $)

Let a=1, b=2, c= -1, d=0 (then then $\displaystyle \frac{\partial u}{\partial \beta} = 1 $ )

etc. ?

Are there any values that would somehow lead me to the incorrect solution?