MATH 046 Lecture Notes - Lecture 7: Integrating Factor, Product Rule
Document Summary
The integrating factor for y + p(x)y = q(x) is = e(cid:2) p(x)dx. Example(cid:1) (cid:1)for the(cid:1) equation (1) find the integrating factor (2) solve the equation using the integrating factor. y + 2y = ex. Answer(cid:1) to(cid:1) example (1) p(x) = 2. (cid:2) p(x) = 2x and the integrating factor is e2x. (2) we multiple the equation with e2x and get e2xy + 2e2xy = e3x. Integrating, we get or (e2xy) = e3x e2xy = e3x + c. For(cid:1) the(cid:1) equation y + 4y = e 4x (1) find the integrating factor (2) solve the equation using the integrating factor. Answer(cid:1) to(cid:1) in(cid:1) class(cid:1) exercise (1) p(x) = 4. (cid:2) p(x)dx = 4x and the integrating factor is e4x. (2) we multiple the equation with e4x and get e4xy + 4e4xy = 1. Using both the product rule and chain rule, this is the same as the following.
