- #1

economicsnerd

- 269

- 24

*nothing*about DEs, so this may be a silly question.

I'm given some time varying [itex](x_t)_t[/itex] and a constant [itex]r[/itex], and I want to solve the equation [itex]u_t = rx_t + \dot u_t[/itex] for [itex]u[/itex].

What I know so far is that (solving the homogeneous equation) if [itex]\bar u[/itex] is some particular solution, then any [itex]u[/itex] is a solution iff it takes the form [itex]u_t=\bar u_t + \alpha e^{rt}[/itex] for some constant [itex]\alpha[/itex].

I'm wondering whether there's a brute force way of finding some [itex]\bar u[/itex]. Anywhere I've looked suggests the "method of undetermined coefficients", but I know it's not useful in my setting. Is there a formula I can blindly apply to get a particular solution? I'm happy to assume [itex]x[/itex] as well behaved as needed, and I'm happy to have my formula be some horrible definite integral I can't compute.