# Linear equations - use of integrating factor

1. Nov 24, 2012

### 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 = e2x

In this case, we begin multiplying both sides by e5x. This gives

e5x$\frac{dy}{dx}$ + y5e5x = e2x.e5x = e7x

We now find that the left-hand-side is the derivative of y.e5x.

$\frac{d}{dx}$(y.e5x) = e7x

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Can someone please clarify how the left hand side is the derivative of y.e5x.

I get that y5e5x is the derivative of y.e5x, but what about the first part of the left-hand-side. What happens to the e5x$\frac{dy}{dx}$

2. Nov 24, 2012

### lurflurf

The product rule is

(uv)'=u'v+uv'

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

You have taken y'=0

3. Nov 24, 2012

### nufc365

Thanks alot lurflurf. It seems so obvious now!

I was treating y as a constant. doh