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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