Here’s a curve. Is y a function of x, or the other way around? In differential form, P dx + Q dy = 0, neither variable is in charge, and solutions trace out whole curves.
Rewrite dy/dx = (a fraction) as P dx + Q dy = 0 without losing the minus sign.
Use partial derivatives and the two-variable chain rule to show that level sets F(x, y) = C are solution curves.
Split a tilted ellipse into two solutions where its tangent turns vertical, and pick the arc through your point.
Chapters
0:00 The question
0:27 Why the two forms agree
0:59 Either variable, any letters
1:35 Clearing a fraction
2:25 Partials and the chain rule
3:20 Level sets
3:55 Example 1: Example one, the tilted ellipse
6:01 Common mistakes
6:19 Practice 1: Check yourself, problem one
7:04 Practice 2: Check yourself, problem two
7:51 Practice 3: Check yourself, problem three
8:40 Recap
9:06 Next time
This is part 17 of Math 33B: Differential Equations. Watch the playlist in order, or jump in wherever you need help.
Free lessons, notes and practice:
Follows the UCLA MATH 33B syllabus. Original AI-generated lesson (AI narration and animation). Not affiliated with any university or instructor.
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