# Y' = [(xy)^2 – xy]/x^2

## 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.

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!

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mfb
Mentor
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.

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:
mfb
Mentor
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.

Alright, thanks! :)