Comparing fixed and variable mortgage rates 
Is a variable rate mortgage with $c =.09 +.001t$ for $20$ years better or worse than a fixed rate of $10\%$?

Let $c$ be the interest rate, $y_0$ be the loan and $s$ be the payment. The differential equation is $\frac{dy}{dt}=cy+s$, $y(0)=y_0$. The solution is $y=y_0e^{ct}+\frac sc(e^{ct}-1)$. But $y=0$ at the end, and we want to compare $s$. So we rewrite it as $s=\frac{-cy_0e^{ct}}{e^{ct}-1}$. When $c$ is fixed, it is easy: $s=\frac{-0.1e^2}{e^2-1}y_0$. However, when $c=0.09+0.001t$, $s=\int_0^{20}\frac{(-0.09+0.001t)y_0e^{0.09t+0.001t^2}}{e^{0.09t+0.001t^2}-1}dt$, which is very hard to compute. How do we compare the two rates?
 A: If, as you wrote, the differential equation is $$\frac{dy}{dt}=(a+b t)y+s\qquad \qquad (y(0)=y_0)$$ the solution is very different if $b=0$ or not.
If $b=0$, as you wrote, $$y=y_0\,e^{at}+\frac s a(e^{at}-1)\implies s_1=-\frac{a e^{a t}}{e^{a t}-1}y_0$$ However, the problem becomes much more complex if $b\neq 0$; in such a case,  the integration of the differential equation leads to $$y= e^{(a +\frac{b t}{2})t}\,y_0+s \,\sqrt{\frac{\pi }{2b}}   e^{\frac{(a+b t)^2}{2 b}}
   \left(\text{erf}\left(\frac{a+b t}{\sqrt{2b}
   }\right)-\text{erf}\left(\frac{a}{\sqrt{2b} }\right)\right)$$ which leads to $$s_2=-\frac{\sqrt{\frac{2b}{\pi }} \,  e^{-\frac{a^2}{2
   b}}}{\text{erf}\left(\frac{a+b t}{\sqrt{2b} }\right)-\text{erf}\left(\frac{a}{\sqrt{2b}
   }\right)}\,y_0$$ where appears the error function.
Now, let us use your numbers for $t=20$.


*

*for $s_1$, $a=\frac 1{10}$ leads to $$s_1=-\frac{e^2}{10(e^2-1)}y_0\approx -0.115652 \,y_0$$

*for $s_2$, $a=\frac 9{100}$, $b=\frac 1{1000}$ leads to $$s_2=\frac{y_0}{10 \,e^{81/20} \sqrt{5 \pi } \left(\text{erf}\left(\frac{9}{2
   \sqrt{5}}\right)-\text{erf}\left(\frac{11}{2 \sqrt{5}}\right)\right)}\approx -0.112074\, y_0$$ which makes $$\frac{s_2}{s_1}\approx 0.969$$

