• Support PF! Buy your school textbooks, materials and every day products Here!

Y' = [(xy)^2 – xy]/x^2

  • Thread starter s3a
  • Start date
  • #1
s3a
800
8

Homework Statement


I’m doing a sample exam, and one of the problems is as follows.:
“Solve the differential equation y' = [(xy)^2 – xy]/x^2.

Hint: Try a substitution.”

The teacher’s solution to this problem is attached as TheSolution.jpg.

Homework Equations


u = xy

The Attempt at a Solution


After having looked at the solution, I understand how to do this problem, but I’m not too sure I get how I would have known to make that particular substitution, if I didn’t have the solution already provided for me. Am I just supposed to try and visualize a substitution, without any system certainty, that would have a derivative that works well in conjunction with the substitution, such that I end up with a separable differential equation, or is there a nice, systematic way of dealing with this problem?

For example, if this differential equation was homogeneous, y = xv would have been a nice substitution that I could rely on to transform the homogeneous, differential equation into a separable one (because that is the systematic way of dealing with homogeneous, differential equations).

Is there a systematic procedure in the teacher’s solution to this problem that I am missing?

Any input would be GREATLY appreciated!
 

Attachments

Answers and Replies

  • #2
34,376
10,466
Am I just supposed to try and visualize a substitution, without any system certainty, that would have a derivative that works well in conjunction with the substitution, such that I end up with a separable differential equation
That's the basic idea. With experience, the ratio of useful to useless substitution attempts will increase.
 
  • #3
s3a
800
8
Thanks for the reply.

Just to confirm, for these kinds of problems, is the substitution always supposed to yield a separable, differential equation, or could it be any other kind of differential equation that I am able to systematically solve?
 
Last edited:
  • #4
34,376
10,466
It can give any other equation type - it can get easy, it can need another substitution to solve, and it can even be worse than the original equation.
 
  • #5
s3a
800
8
Alright, thanks! :)
 

Related Threads on Y' = [(xy)^2 – xy]/x^2

  • Last Post
Replies
5
Views
2K
Replies
3
Views
11K
Replies
3
Views
1K
Replies
1
Views
7K
Replies
2
Views
1K
Replies
2
Views
4K
Replies
5
Views
7K
  • Last Post
Replies
5
Views
4K
  • Last Post
Replies
6
Views
1K
Replies
5
Views
3K
Top