Linear equations - use of integrating factor

[nufc365]

Hi, I am doing self study and have hit a snag in the road. Can someone please clarfiy this for me. I am reading K.A. Stroud's Engineering Mathematics which so far has been great.Consider the equation \frac{dy}{dx} + 5y = e^2x

In this case, we begin multiplying both sides by e^5x. This gives

e^5x\frac{dy}{dx} + y5e^5x = e^2x.e^5x = e^7x

We now find that the left-hand-side is the derivative of y.e^5x.

\frac{d}{dx}(y.e^5x) = e^7x____________________

Can someone please clarify how the left hand side is the derivative of y.e^5x.

I get that y5e^5x is the derivative of y.e^5x, but what about the first part of the left-hand-side. What happens to the e^5x\frac{dy}{dx}

 

[lurflurf]

The product rule is

(uv)'=u'v+uv'

here u=y v=e^(5x)

You have taken y'=0

 

[nufc365]

Thanks a lot lurflurf. It seems so obvious now!

I was treating y as a constant. doh