First Order Linear Nonhomogenous Differential Equation

[IBY]

A question, if you have a nonhomogenous first order differential equation, can it be solved by using the usual nonhomogenous way, with the arbitrary constants, instead of using the integrating factor?

 

[HerpaDerp]

You can use variation of parameters or method of superposition

 

[HallsofIvy]

The question did NOT state that the equation was linear and both "variation of parameters" and the "method of superposition" require that the equation be linear.

 

[IBY]

So, let's take the following:
$$Ay'+By=C$$

I know its solution is: $$\frac{C}{B}(1-e^{-t\frac{B}{A}})$$

For the homogenous part:
$$Ay'+By=0$$
Characteristic equation is:
$$Ak+B=0$$
$$k=-\frac{B}{A}$$

So the solution for the homogenous part is:
$$y=e^{-t\frac{B}{A}}$$

Now, from what I have read about variation of parameters, there is a function v such that:
$$Y=vy$$
$$Y'=v'y+vy'$$

So, would that be:
$$vBe^{-t\frac{B}{A}}=Y$$
$$v'Be^{-t\frac{B}{A}}-v\frac{B^2}{A}e^{-t\frac{B}{A}}=Y'$$

Err... Correct?