sanctus Posted March 5, 2007 Report Posted March 5, 2007 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? Quote
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