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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?

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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 $y = c_1e^{-2x}+2$.

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    @MichaelHardy thanks for the comment, fixed it.2012-10-05
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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.

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    That characteristic equation is wron$g$...2012-10-21
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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.