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I'm trying to verify that the following differential equation:

 

[math]\partial_tM+n^i\partial_iM=-n^j\partial_j(\Psi+\Phi)[/math]

 

has the following general solution with initial condition [math]M(t_{in},\vec{x},\vec{n})[/math]

 

[math]M(t,\vec{x},\vec{n})=M(t_{in},\vec{x}-\vec{n}(t-t_{in}),\vec{n})-\int_{t_{in}}^t dt' n^i\partial_i(\Psi+\Phi)(t',\vec{x}-\vec{n}(t-t'))[/math]

 

One can easily see that

[math]\partial_tM(t_{in},\vec{x}-\vec{n}(t-t_{in}),\vec{n})=-n^i\partial_iM(t_{in},\vec{x}-\vec{n}(t-t_{in}),\vec{n})[/math]

[math]\partial_t(\Psi+\Phi)(t',\vec{x}-\vec{n}(t-t'))=-n^i\partial_i(\Psi+\Phi)(t',\vec{x}-\vec{n}(t-t'))[/math]

 

So, replacing the general solution in the lefthand side of the differential equation, I get zero instead of

[math]-n^j\partial_j(\Psi+\Phi)[/math]

 

This is not logic because the general solution is just the sum of a solution of the homogene equation (M(t_{in},...))and a particular solution (the integral)...

Where did I make a mistake?

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