A First Order Nonlinear ODE Is there a closed form solution for this first order ODE of $x(t)$ where $a(t)$ is as smooth as needed?
$$x'(t)=\frac{a(t)}{x(t)}+b$$
Here, $b$ is a constant. But what if $b$ is also a function of $t$? But I do not want a series expansion formula with recursion equations resulting from the Frobenius method.

This belongs to an Abel equation of the second kind.
In fact all Abel equation of the second kind can be transformed into Abel equation of the first kind.
Let $x(t)=\dfrac{1}{y(t)}$ ,
Then $x'(t)=-\dfrac{y'(t)}{(y(t))^2}$
$\therefore-\dfrac{y'(t)}{(y(t))^2}=a(t)y(t)+b(t)$
$y'(t)=-a(t)(y(t))^3-b(t)(y(t))^2$
Please follow the method in http://www.hindawi.com/journals/ijmms/2011/387429/#sec2
