# Thread: Simple question for linear first-order differential equation. I can't move on!

1. ## Simple question for linear first-order differential equation. I can't move on!

Could someone please tell me how to do this simple question so that I can move on to more complex stuff?

y' - y = 6x; y(x) = 4

y(x) = ?

What I DO get is that I need an integrating factor since this is a first-order linear differential equation. I obtain the integrating factor by recognizing that the first-order linear differential equation has the form y' + p(x)y = q(x) and that the integrating factor has the form I(x) = exp(integral(p(x) dx)). Then, there is an equation that I think I am supposed to memorize (since I don't see how to get from the previous step to it - if you can show me that would be good): d(yI)/dx = Iq(x). The previous step is I(x)y' + p(x)I(x)y = I(x)q(x) which is basically taking the first-order linear differential equation and multiplying it by the integrating factor.

What I DON'T get is, I'm supposed to "integrate both sides of this last equation with respect to x, and then solve the resulting equation for y" but I can't seem to do that in this case.

Any help would be GREATLY appreciated!

2. ## Re: Simple question for linear first-order differential equation. I can't move on!

Originally Posted by s3a
Could someone please tell me how to do this simple question so that I can move on to more complex stuff?

y' - y = 6x; y(x) = 4

y(x) = ?

What I DO get is that I need an integrating factor since this is a first-order linear differential equation. I obtain the integrating factor by recognizing that the first-order linear differential equation has the form y' + p(x)y = q(x) and that the integrating factor has the form I(x) = exp(integral(p(x) dx)). Then, there is an equation that I think I am supposed to memorize (since I don't see how to get from the previous step to it - if you can show me that would be good): d(yI)/dx = Iq(x). The previous step is I(x)y' + p(x)I(x)y = I(x)q(x) which is basically taking the first-order linear differential equation and multiplying it by the integrating factor.

What I DON'T get is, I'm supposed to "integrate both sides of this last equation with respect to x, and then solve the resulting equation for y" but I can't seem to do that in this case.

Any help would be GREATLY appreciated!
What function did you get for your integrating factor?

$\frac{d}{dx}\left[yI\right] = Iq(x) \Rightarrow yI = \int I q(x) \, dx + C$

and now you're expected to know how to integrate.

3. ## Re: Simple question for linear first-order differential equation. I can't move on!

I actually ended up figuring it out myself. Thanks anyways.