Topics delayed from Week 1.Discussion of answer checks implicit solution ln|y|=2x+c for y'=2y explicit solution y = C exp(2x) for y'=2y Troubles with explicit solutions of y'= 3 sqrt(xy) [1.4-6].Key ExamplesSeparable DE with no equilibrium solutions. Separable DE with finitely many equilibrium solutions. Separable DE with infinitely many equilibrium solutions. The list of answers to a separable DE. Influence of an initial condition to extract just one solution formula from the list. Examples for Midterm 1 problem 2: y'=x+y, y'=x+y^2, y'=x^2+y^2 Example 1: Show that y'=x+y is not separable using TEST I or II (partial derivative tests). Example 2: Find a factorization f=F(x)G(y) for y'=f(x,y), given (1) f(x,y)=2xy+4y+3x+6 [ans: F=x+2, G=2y+3]. (2) f(x,y)=(1-x^2+y^2-x^2y^2)/x^2 [ans: F=(1-x^2)/x^2, G=1+y^2].

Review TopicsDrill: Direction fields. Two Threading Rules. Picard and Peano Theorems. We draw threaded solutions from some dot in the graphic. How do we choose the dots? What do they represent? What does dy/dx=f(x,y), y(x0)=y0 have to do with threaded curves? Drill: Quadrature integral of du/(1+u^2), 2u du/(1+u^2). True and false trig formulas: arctan(tan(theta))=theta [false], tan(arctan(x))=x [true].

Review and Drill 1.4Variables separable method. Discuss remaining exercises 1.4-6,12,18. Problem Notes 1.4 at the web site. Equilibrium solutions and how to find them.

Lecture on Section 1.5Theory of linear DE y'=-p(x)y+q(x). Integrating factor W=e^Q(x), Q(x) = int( p(x),x) (Wy)'/W, the fraction that replaces two-termed expression y'+py.Classification of y'=f(x,y)quadrature [Q], separable [S], linear [L]. Venn diagram of classes Q, S, L. Examples of various types. Test for quadrature (f_y=0) Test for linear (f_y indep of y) Test for not separable (f_y/f depends on x ==> not sep) Finding F and G in a separable equation y'=F(x)G(y)Linear integrating factor method 1.5Application to y'+2y=1 and y'+y=e^x. Examples: Testing linear DE y'=f(x,y) by f_y independent of y. Classifying linear equations and non-linear equations. Picard's theorem implies a linear DE has a unique solution. Main theorem on linear DE and explicit general solution.

- References for linear DE:

Exam 1 date is 17 Feb 1-3pm in WEB 104 or 18 Feb 6:50am in JTB 140. Other exam times were pre-set by agreement at the start of the semester, on an individual basis. The plan was created to provide extra time to write the exam, which is designed for 50 min.

Sample Exam: Exam 1 key from F2009. See also S2009, Exam 1.

Questions on textbook sections 1.3, 1.4.

Review and drill Ch1.

Review and DrillMore about problem 1.5-34 The expected model is x'=1/4-x/16, x(0)=20, using units of millions of cubic feet. The answer is x(t)=4+16 exp(-t/16). Model Derivation Law: x'=input rate - output rate. Definition: concentration == amt/volume. Use of percentages 0.25% concentration means 0.25/100 concentration Superposition for y'+p(x)y=0. Superposition for y'+p(x)y=q(x).

Drill Section 1.5Three linear examples: y'+(1/x)y=1, y'+y=e^x, y'+2y=1. classification: separable, quadrature, linear. Methods for solving first order equations: Linear integrating factor method, Superposition + equilibrium solution for constant-coefficient linear DE Drill: worksheet distributed in class, for the example y' + 2y = 6. Solved in class y'+3y=6, y'+y=e^x, and several homogeneous equations like y'+3y=0, y'+2y=0. Solved for equilibrium solutions in more complicated examples like 2y' + Pi y = e^2.

Drill Section 1.5: Three linear examples: y'+(1/x)y=1, y'+y=x, y'+2y=1. classification: separable, quadrature, linear. Methods for solving first order equations: Linear integrating factor method, Superposition + equilibrium solution for constant-coefficient linear, Method of Quadrature Variables Separable method Equilibrium solutions from G(y)=0 and Non-equilibrium solutions from G(y) nonzero.

Lecture on 2.1Theory of autonomous DE y'=f(y), Picard's theorem and non-crossing of solutions. Direction fields and translation of solutions Constructing Euler's threaded solution diagrams No direction field, just the phase line diagram. Partial fractions. DEFINITION: partial fraction=constant/polynomial with exactly one root THEOREM: P(x)/q(x) = a sum of partial fractions Finding the coefficients. Method of sampling clear fractions, substitute samples, solve for A,B, ... Method of atoms clear fractions, multiply out and match powers, solve for A,B,... Heaviside's cover-up method partially clear fraction, substitute root, find one constant Separation of variable solutions with partial fractions.

- References for 2.1, 2.2:

To date, Murphy and Cox have covered problems 1,2,3 in the exam review sessions on Wed-Thu.