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- Homework Statement
- please see below

- Relevant Equations
- please see below

__Attempt at a solution__To show φ satisfies our PDE, we first solve the substitution for φ

##\mathrm{ln(\phi) = -\frac {1} {2} \int u dx}##

which gives

##\mathrm{\phi = e^{-\frac {1} {2} \int u dx} }##

and plug it into our PDE, which simplifies to

##\mathrm{\frac {\partial } {\partial t} - \frac {\partial} {\partial x} u } = 0 ##

we use the substitution φ to show the following

##\mathrm{\frac {\partial } {\partial t} (\frac {-2 \frac {\partial} {\partial x} \phi} {\phi}) - {\frac {\partial ^2} {\partial x^2} (\frac {-2 \frac {\partial} {\partial x} \phi} {\phi} = 0 ##

so φ solves our PDE. To show u solves the Burger's equation with viscosity, we substitute u into the heat equation and simplify

which is where I am stuck. It seems like this equation does not simplify to anything useful. What I am getting is the following.

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