Solving a Differential Equation with Substitution | y' = [(xy)^2 – xy]/x^2

[s3a]

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!

 

[mfb]

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.

 

[s3a]

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?

 

[mfb]

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.

 

[s3a]

Alright, thanks! :)