I am struggling to work out what the force of interest for simple interest is when using differential equations. I know that it is $\delta=\frac{r}{1+rt}$ where $r$ is the interest rate, but when I try to find the force of interest using differential equations I get a different answer.
Simple interest has accumulation function $a(t)=1+rt$. By definition the force of interest $\delta(t)$ is $\delta(t)=\frac{da(t)/dt}{a(t)}$, so clearly we have that $\delta(t)=r/(1+rt)$.
However, $\frac{da(t)}{dt}=a(t)\delta(t)$ is a differential equation, and solving it we have $a(t)=e^{t\delta}$ (we can assume that the constant is $0$ when integrating, and that $\delta$ is constant).
Then we have $1+rt=e^{t\delta}$, so $\delta=\frac{1}{t}\ln(1+rt)$. Where am I going wrong?