Hello,

I am a little confused by Mathematica's implementation of the variation of parameters method in DSolve. Consider solving a general inhomogenous function:

DSolve[f''[x] + f[x] == g[x], f[x],x]

The provided solution is:

f[x] = Exp[x]*C[1] + Exp[-x]*C[2] + Exp[-x]*(Exp[2*x]* Integrate[1/2*Exp[-k]*g[k], {k, 1, x}] + Integrate[-1/2* Exp[k]*g[k], {k, 1, x}])

My question is this: why are the limits of integration for the particular solution 1 to x?

In the general expression for variation of parameters, the integral is indefinite. By applying the second fundamental theorem of calculus, the indefinite integral can be converted to a definite integral from x0 to x

*provided* the integrand is continuous and at the lower limit of integration (x0), the integrand evaluates to zero. However, Mathematica does not seem to enforce either of these requirements, so I doubt this is the reason the limits are as given. Is there a Green's function arguement that would justify these limits for the general function g?

Thank you,

Anna