Ordinary Differential Equation Strategy In one of the examples in the Differential Equations for Dummies Workbook (Holzner), you are asked to use an integrating factor to solve 
$$ \frac{dy}{dx} +2y =4 $$
My question is, is this the most efficient way to solve it? Can't I also solve it by separating the $y$, turning the equation into $\frac{1}{2-y}\frac{dy}{dx} = 2$. Are there other ways? How do you quickly determine what will be the quickest way?
 A: The quickest way to solve this is to note $\lambda+2=0$ gives $\lambda=-2$ hence $y_h = e^{-2x}$ and eyeballin-it shows $y_p = 2$ hence $y = c_1e^{-2x}+2$. 
A: Another to solve it would be to use the characterstic equation which is what the above answer used:
z^2 + 2z - 4 = 0.  
You can factor this equation and the general solution is would by using the general formula provided in any ODE text.
However, integrating factors do not require you to remember a formula and in most situations this technique is more elegant and quick.
A: You can separate variables, thus:
$$
\frac{dy}{dx} = 4-2y
$$
$$
\frac{dy}{2-y} = 2\,dx
$$
$$
-\log|2-y| = 2x+\text{constant}
$$
$$
|2-y|=e^{-2x-\text{constant}} = (e^{-2x}\cdot(\text{positive constant}))
$$
$$
2-y = e^{-2x}\cdot\text{constant}
$$
$$
y = 2-(e^{-2x}\cdot\text{constant})
$$
As for integrating factors, notice that you have $y'+2y$ and after multiplying by some factor---call it $w$---you have $wy'+(2w)y$ and you want $wy'+w'y$, so that it becomes $(wy)'$.  So you need $w'=2w$.  That's a differential equation, one of whose solutions is $w=e^{2x}$.  And you only need one of its solutions.
