Drill on 1.5 Problems There are two special methods for solving y'+py=q If p,q are constant then use the SUPERPOSITION METHOD y = y_p + y_h y_p = an equilibrium solution (set y'=0, solve for y) y_h = constant divided by the integrating factor If one of p or q depends on x, then use the STANDARD METHOD Replace the LHS, which is y'+p(x)y, by the integrating factor quotient (Wy)'/W, where W=exp(int p(x)dx)) is the integrating factor. Cross-multiply by W to clear fractions. Then apply the method of quadrature.

General Verhulst DESolving y'=(a-by)y by a substitution Let u=y/(a-by). Then substitution into the DE gives u'=au Solve u'=au to get u=u0 exp(ax). Back-substitute u(x) into u=y/(a-by), then solve for y. Solving y'=(a-by)y by partial fractions Divide the DE by (a-by)y Apply the method of quadrature. Find the constants in the partial fractions on the left. Integrate to get the answer a y0 y(x) = -------------------------- b y0 + (a - b y0) exp(-ax) where y0=y(0)=initial population size.

Lecture on 2.2:Theory 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 is needed to draw solution curves We throw out the threaded solution rule used in chapter 1, replace it by two rules from calculus and a theorem: 1. If y'(x)>0, then y(x) increases. 2. If y'(x)<0, then y(x) decreases. THEOREM. For y'=f(y), a threaded solution starting with y'(0)>0 must satisfy y'(x)>0 for x>0. A similar result holds for y'(0)<0. Definition: phase line diagram, phase diagram, Calculus tools: f'(x) pos/neg ==> increasing/decreasing DE tool: solutions don't cross Maple tools for production work. Stability theory of autonomous DE y'=f(y) Stability of equilibrium solutions. Stable and unstable classification of equilibrium solutions. funnel, spout, node, How to construct Phase line diagrams How to make a phase diagram graphic Inventing a graph window Invention of the grid points Using the phase line diagram to make the graphic calculus tools DE tools

- References for 2.1, 2.2, 2.3. Includes the rabbit problem, partial fraction examples, phase diagram illustrations.

Examples and ApplicationsGrowth-Decay model y'=ky and its algebraic model y=y(0)exp(kx). Pharmokinetics of drug transport [ibuprofen] Pollution models. Three lake pollution model [Erie, Huron, Ontario]. Brine tanks. One-tank model. Two-tank and three-tank models. Recycled brine tanks and limits of chapter 1 methods. Linear cascades and how to solve them. Method 1: Linear integrating factor method. Method 2: Superposition and equilibrium solutions for constant-coefficient y'+py=q. Uses the shortcut for homogeneous DE y'+py=0. Separation of variables The equation y'=7y(y-13), y(0)=17 F(x) = 7, G(y) = y(y-13) Separated form y'/G(y) = F(x) Answer check using the Verhulst solution P(t) = aP_0/(bP_0 + (a-b P_0)exp(-at)) Separation of variables details. Review of partial fractions. Partial fraction details for 1/((u(u-13)) = A/u + B/(u-13)

- References for 2.1, 2.2, 2.3. Includes the rabbit problem, partial fraction examples, phase diagram illustrations.

Drill and ReviewPhase diagram for y'=y^2(y^2-4) Phase line diagram Threaded curves Labels: stable, unstable, funnel, spout, node Phase line diagrams. Phase diagram.

Newton's force and friction modelsIsaac Newton ascent and descent kinematic models. Free fall with no air resistance F=0. Linear air resistance models F=kx'. Non-linear air resistance models F=k|x'|^2.

Problem notes for 2.3-10. 2.3-20 (2.8 K, txt, 28 Jan 2012)

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. Exercise solutions to the problems due in 2.1.

Exam 1 review, questions and examples on exam problems 1,2,3,4,5. Sample Exam: Exam 1 key from F2010. See also S2010, exam 1. Lecture on midterm 1 problems 4,5. Lecture on 2.2-10,18. Maple Lab 2, problem 1 details [maple L2.1].