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Given an inhomogeneous ODE of the form $$a_{n}(x)y^{(n)}(x)+a_{n-1}(x)y^{(n-1)}(x)+\cdots +a_{2}(x)y''(x)+a_{1}(x)y'(x)+a_{0}(x)y(x)=f(x)$$ where ##y^{(n)}(x)\equiv \frac{d^{n}y(x)}{dx^{n}}##, why is the function ##f(x)## on the right hand side referred to as a "source term" ? In what way does it source left hand side (or the function ##y(x)##)?
The reason I ask as in quantum field theory the Feynman propagator for a scalar field ##\phi (x)## can be derived by considering the Klein-Gordon equation with a so-called "source term" ##J(x)##, i.e. $$\left(\Box +m^{2}\right)\phi (x)=J(x)$$ where ##\Box\equiv\eta^{\mu\nu}\frac{\partial^{2}}{\partial x^{\mu}\partial x^{\nu}}##. I'm unsure as to how this should be interpreted physically, or whether it is just a useful mathematical trick that bears no physical interpretation?! (I understand the derivation of the Feynman propagator for a scalar field from this, but I'm unsure how to motivate the introduction of the source term ##J(x)## on the right hand side of the K-G equation?!)
N.B. I originally posted this in the differential equations forum, but it was suggested that perhaps it would be more suited in this forum, given what the question is relating to.
The reason I ask as in quantum field theory the Feynman propagator for a scalar field ##\phi (x)## can be derived by considering the Klein-Gordon equation with a so-called "source term" ##J(x)##, i.e. $$\left(\Box +m^{2}\right)\phi (x)=J(x)$$ where ##\Box\equiv\eta^{\mu\nu}\frac{\partial^{2}}{\partial x^{\mu}\partial x^{\nu}}##. I'm unsure as to how this should be interpreted physically, or whether it is just a useful mathematical trick that bears no physical interpretation?! (I understand the derivation of the Feynman propagator for a scalar field from this, but I'm unsure how to motivate the introduction of the source term ##J(x)## on the right hand side of the K-G equation?!)
N.B. I originally posted this in the differential equations forum, but it was suggested that perhaps it would be more suited in this forum, given what the question is relating to.