Math 4335 HomeworkInstructions:•Staple or bindall pages together.DO NOTdog ear pages as a method to bind.•Hand-drawn sketches should be neat, clear, of reasonable size, with axis and tick marks...

Math 4335 Homework Instructions: • Staple or bind all pages together. DO NOT dog ear pages as a method to bind. • Hand-drawn sketches should be neat, clear, of reasonable size, with axis and tick marks appropriately labeled. All ?gures should have a short caption explaining what they show and describe. • Please label each problem and write clearly. If your work can not be read, or the problem not found, it will not be graded. 1 Review of ODEs • Reading (Britton): Sec. B.1.2 has a ridiculously short presentation of separation of variables. Solve the following ?rst-order ODEs 1. ' 2 u =(t +2), u(3)=2 2. 2 t +4 ' u = 2 u +4 3. ' 2 u =cos(t)u 4. ' u =u(u-1), u(0)=u , (Use partial fractions.) 0 5. ' u +4u=t, u(0)=1 6. ' t tu +u =e , u(1)=1st 2 Graphical Analysis of 1 -order ODEs • Reading (Britton): Sec. B.1.1 has a ridiculously short presentation of the phase line. 1. We considered the ODE below as a simple model for the population of a ?sh tank ' u = (k -k )u+(l -l )=ku+l. 2 1 2 1 “Real” ?sh require u > 0 such that the equilibrium point u =-l/k requires either k <> 0 or k > 0, l <>