MATH 046 Lecture 3: First Order Differential Equations Notes
Document Summary
(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) first(cid:1) order(cid:1) differential(cid:1) equations. Here are the things we will learn today: (1) particular solution and general solution. (2) initial value problems for rst order equations. (3) classi cation of rst order equations. (cid:1) (cid:1) (cid:1) particular(cid:1) solution(cid:1) and(cid:1) general(cid:1) solution. We(cid:1)also(cid:1)observed(cid:1)that(cid:1)a(cid:1)di erential(cid:1)equation(cid:1)might(cid:1)have(cid:1)many(cid:1)solutions. (cid:1) to(cid:1)emphasis(cid:1)this(cid:1) fact,(cid:1) we(cid:1) distinguish(cid:1) between(cid:1) particular(cid:1) solution(cid:1) and(cid:1) general(cid:1) solution. A(cid:1) particular(cid:1) solution(cid:1) is(cid:1) one(cid:1) of(cid:1) the(cid:1) solution(cid:1) (any(cid:1) solution)(cid:1) of(cid:1) a(cid:1) di erential(cid:1) equation. (cid:1) (x)(cid:1) (cid:1)y(x)(cid:1)=(cid:1)0. (1) y(x) = e (2) y(x) = 2e (3) y(x) = ce x is a particular solution. x is also a particular solution. x is the general solution, where c can be any number. (x) = ce x and x = 0. In(cid:1) particular,(cid:1) c(cid:1)=(cid:1)1(cid:1) and(cid:1) c(cid:1)=(cid:1)2(cid:1) give(cid:1) the(cid:1) above(cid:1) two(cid:1) particular(cid:1) solutions. (x) y(x) = 0 y where(cid:1) c1(cid:1) and(cid:1) c2(cid:1) can(cid:1) be(cid:1) any(cid:1) numbers.
