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Hey I'm currently stuck on this question. I don't think its in the form of an integrating factor, but I'm also not sure if its separable?

$$\frac {dy}{dx}=-3y+2e^{-x}$$

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The general form for an equation solvable by integrating factor is $$\frac{dy}{dx}+P(x)y=Q(x).$$

In your case I see $$\frac{dy}{dx}+3y=2e^{-x}.$$

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  • $\begingroup$ But the second part for P(x)y don't we need an x term? Like, I'm not sure how to find the integrating factor here. $\endgroup$ – user176330 Sep 19 '14 at 5:46
  • $\begingroup$ How about $P(x)=3x^0$. There is your $x$ term. ;). Not to be a smarty pants, just think of $P(x)$ as being $P(x)=3$. $\endgroup$ – J. W. Perry Sep 19 '14 at 5:48
  • $\begingroup$ Oh i see that now! Could you tell me if I'm correct, where the integrating factor is: $$e^{3x+c}$$ or simplified to be $$e^{3x}$$ $\endgroup$ – user176330 Sep 19 '14 at 5:58
  • $\begingroup$ If your have $\mu(x)=e^{\int 3\, dx }$ then you are correct. $\endgroup$ – J. W. Perry Sep 19 '14 at 6:01
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    $\begingroup$ user176330 this kind of equations is easily solved by using the superposition principle, i always found the integration factor method a bit too specific. if you are interested in a more general/intuitive way of solving DE, comment below, btw +1 for @Perry $\endgroup$ – Francesco Alem. Sep 19 '14 at 6:06

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