First-Order Equations: don't understand a simplification step

[Kavorka]

There is a specific step when solving first-order equations introduced in my textbook when describing the process for solving with an integrating factor, and it has been popping up everywhere - I do not understand how it works and the book doesn't explain it well.

Example 1:

x(dy/dx) + y = sin(x) / x

becomes

d/dx(xy) = sin(x) / x

Example 2:

e^-x (dy/dx) - y(e^-x) = e^(-4x/3)

becomes

d/dx(y(e^-x)) = e^(-4x/3)I'm having trouble rationalizing how this step is performed. Thanks for any help!

 

[Simon Bridge]

Try a different lesson on integrating factors...
http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx
... then attempt to use them on your examples and get back to us.

 

[Mark44]

There is a specific step when solving first-order equations introduced in my textbook when describing the process for solving with an integrating factor, and it has been popping up everywhere - I do not understand how it works and the book doesn't explain it well.

Example 1:

x(dy/dx) + y = sin(x) / x

becomes

d/dx(xy) = sin(x) / x

Example 2:

e^-x (dy/dx) - y(e^-x) = e^(-4x/3)

becomes

d/dx(y(e^-x)) = e^(-4x/3)I'm having trouble rationalizing how this step is performed. Thanks for any help!

In both examples they are recognizing that the left side is the derivative, with respect to x, of a product. In the first example, the product is xy. In the second, the product is ye^-x.