There is a second order nonhomogeneous equation of motion with nonzero initial condition given at ##t=-\infty##:(adsbygoogle = window.adsbygoogle || []).push({});

##D^2 y(x)=f(x)## with ##y(-\infty)=e^{-i x}##

where I have used the shorthand notation ##D^2## for the full differential operator. Also I have the two solutions ##y_1(x)## and ##y_2(x)## to the homogeneous equation with ##y_1(x \to -\infty) \approx e^{-ix}## and ##y_2(x \to -\infty) \approx e^{ix}##. So how do I construct the particular solution using ##y_1(x)## and ##y_2(x)##?

I know Green's function can be constructed using the two homogeneous solutions. So the naive solution I got is

##y(x)=y_1(x)+\int^x_{-\infty} dx' G(x,x')f(x')## with

##G(x,x')=\frac{y_1(x)y_2(x')-y_2(x)y_1(x')}{W(y_2(x'),y_1(x'))}##.

Is it wrong? since I get some ridiculous result (divergence) when I use this solution to do calculation, because the lower bound is ##-\infty##.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A question about Green's function

Loading...

Similar Threads - question Green's function | Date |
---|---|

I A few questions about Green's Functions... | Feb 12, 2017 |

I A question about boundary conditions in Green's functions | Dec 20, 2016 |

I A somewhat conceptual question about Green's functions | Dec 17, 2016 |

Question on why the book claimed Green's function =< 0. | Aug 9, 2010 |

Question about function defined in a region using Green's identities. | Jul 31, 2010 |

**Physics Forums - The Fusion of Science and Community**