In studying a reflection-transmission problem involving exotic materials, I have come across the following linear first-order differential equation:

$\begin{array}{}\text{(1)}& A\frac{\mathrm{\partial}}{\mathrm{\partial}t}g(t)+Bg(t)=f(t),\end{array}$

where A and B are constants, g(t) is associated with the reflected wave, and f(t) is a (finite) driving function associated with the incident wave. Both A and B may be positive or negative. I am interested in the behavior of the solution in the limit that $A\to 0$

In studying a reflection-transmission problem involving exotic materials, I have come across the following linear first-order differential equation:A∂∂tg(t)+Bg(t)=f(t),(1)where A and B are constants, g(t) is associated with the reflected wave, and f(t) is a (finite) driving function associated with the incident wave. Both A and B may be positive or negative. I am interested in the behavior of the solution in the limit that A\rightarrow0.

I know there is an exact solution to Eq. (1), which is

$g(t)=C{e}^{-Bt/A}+\frac{1}{A}{\int}_{-\mathrm{\infty}}^{t}{e}^{-B(t-{t}^{\prime})/A}f({t}^{\prime})d{t}^{\prime},$

where C=0 because g(t)=0 if f(t)=0. However, I do not understand how this exact solution reduces to the case where A=0, which is $g(t)={B}^{-1}f(t)$. Any insight would be greatly appreciated.

I've seen a lot of documents discussing asymptotic analyses of linear differential equations, but they all start with second-order equations. Is this because there is inherently problematic with first-order?

$\begin{array}{}\text{(1)}& A\frac{\mathrm{\partial}}{\mathrm{\partial}t}g(t)+Bg(t)=f(t),\end{array}$

where A and B are constants, g(t) is associated with the reflected wave, and f(t) is a (finite) driving function associated with the incident wave. Both A and B may be positive or negative. I am interested in the behavior of the solution in the limit that $A\to 0$

In studying a reflection-transmission problem involving exotic materials, I have come across the following linear first-order differential equation:A∂∂tg(t)+Bg(t)=f(t),(1)where A and B are constants, g(t) is associated with the reflected wave, and f(t) is a (finite) driving function associated with the incident wave. Both A and B may be positive or negative. I am interested in the behavior of the solution in the limit that A\rightarrow0.

I know there is an exact solution to Eq. (1), which is

$g(t)=C{e}^{-Bt/A}+\frac{1}{A}{\int}_{-\mathrm{\infty}}^{t}{e}^{-B(t-{t}^{\prime})/A}f({t}^{\prime})d{t}^{\prime},$

where C=0 because g(t)=0 if f(t)=0. However, I do not understand how this exact solution reduces to the case where A=0, which is $g(t)={B}^{-1}f(t)$. Any insight would be greatly appreciated.

I've seen a lot of documents discussing asymptotic analyses of linear differential equations, but they all start with second-order equations. Is this because there is inherently problematic with first-order?