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!

Thanks in advance!

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

Quote:

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!

Thanks in advance!

What function did you get for your integrating factor?

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

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

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