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

  • Thread starter Kavorka
  • Start date
  • #1
95
0

Main Question or Discussion Point

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!
 

Answers and Replies

  • #2
Simon Bridge
Science Advisor
Homework Helper
17,847
1,644
  • #3
33,279
4,986
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.
 

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

  • Last Post
Replies
1
Views
855
Replies
7
Views
902
Replies
1
Views
3K
  • Last Post
Replies
7
Views
2K
Replies
1
Views
2K
Replies
2
Views
1K
Replies
4
Views
776
Replies
11
Views
985
Replies
3
Views
1K
  • Last Post
Replies
4
Views
2K
Top