Solve non homogeneous rational equation Is there a close form solution for an equation of this type
$\frac{dx}{dt}=\frac{a}{b+c\cdot t}-d\cdot x$
with $a, b, c, d$ positive constants ?

$$\dfrac{dx}{dt} + d \cdot x = \dfrac{a}{b+ct}$$
$$\dfrac{dx}{dt} \cdot e^{d \cdot t} + d \cdot x \cdot e^{d \cdot t} = \dfrac{a \cdot e^{d \cdot t} }{b+ct}$$
$$\dfrac{d \left(x \cdot e^{d \cdot t} \right)}{dt} = \dfrac{a \cdot e^{d \cdot t} }{b+ct}$$
Hence,
$$x(t) \cdot e^{d \cdot t} - x(0) = \int_0^t \dfrac{a \cdot e^{d \cdot s} }{b+cs} ds \implies x(t) = x(0) e^{-d \cdot t} + \int_0^t \dfrac{a \cdot e^{d \cdot (s-t)} }{b+cs} ds$$
