$\frac {dc}{dx} = b + b_1 x - a c$. Find $ c $ as a function of $ x $ if $ c=0 $ when $ x=0 $

Question

Manufacturing and marketing costs $ c $ are related to the number of items $ x $ by the relation:-

$\frac {dc}{dx} = b + b_1 x - a c$ ($a , b_1 , b$ are constants)

Find $ c $ as a function of $ x $ if $ c = 0 $ when $ x = 0 $.

Answer 1

Hint:

Notice that

$$\frac{d}{dx}c(x)+ac(x)=e^{-ax}\frac{d}{dx}(c(x)e^{ax})$$ so that

$$\frac{d}{dx}(c(x)e^{ax})=(b+b_1x)e^{ax}$$ and $$c(x)e^{ax}-c(0)=\int_0^x(b+b_1z)e^{az}dz.$$

The integral can be evaluated by parts.

Answer 2

As a continuation to @Yves Daoust answer

$$c(x)e^{ax} = \frac {be^{ax} + b_1e^{ax}(x-1)}{a} + k$$ $$c(x) = \frac {b + b_1x - b_1}{a} + k $$