Worksheet 1-3:
Existence-Exact equations
Supplemental Instruction
Iowa State University
Leader:
Course:
Instructor:
Date:
Zaynab Diallo
Math 267
D’Alessandro
1/21/16
Existence and uniqueness of solution
Let 𝑦 ′ = 𝑥𝑦 2 , 𝑦(0) = 1
Does this initial value problem have a (unique) solution? Please Justify?
Hint: continuity of f(x,y) and f’(x,y)
Exact Equations
𝑑𝑦 3𝑥 2 + 4𝑥 + 2
=
, 𝑦(0) = −1
𝑑𝑥
2(𝑦 − 1)
1) Rewrite this equation in its form 𝑵𝒅𝒚 + 𝑴𝒅𝒙 = 𝟎 𝑜𝑟
2) Identify 𝑵(𝒙, 𝒚)𝒂𝒏𝒅 𝑴(𝒙, 𝒚)
3) Is the differential equation an exact equation?
Hint: verify that
𝒅𝑵
𝒅𝒙
=
𝒅𝑴
𝒅𝒚
𝑵𝒅𝒚
𝒅𝒙
+𝑴=𝟎
Problem 1
Find the solution of these differential equations using separable technique and linear genera
form
1) 𝑦 ′ = 𝑥 2 𝑦