# Ma316-001-F10 Day-by-Day Syllabus

This syllabus is for Math 316, Fall 2010, taught by Gavin LaRose. Homework assignments are also available.
Please Note: the schedule and assignments listed here are subject to change. They should not change within a week of due dates, but may change further in advance.

Mon Tue Wed Thu Fri
Sep 8: Intro., 1.1-1.3 Sep 9

Note: S1.1: meaning of differential equation, equilibrium solution, and how direction fields are constructed; S1.2: why a DE has a family of solutions, meaning of initial condition and general solution; S1.3: meaning of ordinary and partial differential equation, order, linear and nonlinear differential equation, and solution of a DE; S2.1: how an integrating factor allows solution of a (linear, first order) DE.
(Note: 2.4 also appears 9/17)
Note: S2.2: which eqns are separable, integral curves of a differential eqn, the interval of validity of solns (ex.2,3); S2.3: the derivation of the "rate out" in ex.1,3; S2.4: theorems 2.4.1, 2.4.2, why 2.4.2 does not apply in ex.3, general solutions & nonlinearity.
Sep 14 ----------------
HW 1 Due
Note: S2.5: meaning of autonomous, and of equilibrium solutions and critical points, how to draw a phase line, how this lets us sketch solutions, meaning of asymptotic stability and unstable solutions.
Sep 16 ----------------
Lab 1
Note: S2.4: theorems 2.4.1, 2.4.2, why 2.4.2 does not apply in ex.3, general solutions & nonlinearity; S2.8: how the integral equation (3) is related to (2), what the iteration method is, the four questions we have to answer to establish the theorem, how ex.1 derives the iterates & soln to the equation.
Note: S3.1: form of a linear 2nd order eqn, meaning of homogeneous, how to solve a linear constant-coefficient eqn; S3.2: what a differential operator is, theorem 3.2.1, the three things it says, what superposition is, definition of the Wronskian, how it guarantees a general solution to a linear 2nd order eqn.
Sep 21
Lab 1 Due
Note: S3.3: what Euler's formula is, how to find real-valued solutions when roots to the characteristic eqn are complex.
Sep 23
HW 2 Due
Note: S3.4: the difficulty that arises when a characteristic eqn has repeated roots, the generalization used in ex.1 to find another solution, what reduction of order is.
Note: S3.5: why Y1(t) - Y2(t) is a homogeneous solution, why the general nonhomogeneous solution is c1 y1(t) + c2 y2(t) + Y(t), how the Method of Undetermined Coefficients proceeds, what we guess for Y(t) if g(t) = eat, sin(bt), cos(bt), or a polynomial, the summary on p.180 and table 3.5.1.
Sep 28 Sep 29: Lab 2 Sep 30

Note: S3.6: the basic idea of variation of parameters, why in ex.1 we are able to impose the condition u1'(t) cos(2t) + u2'(t) sin(2t) = 0, the derivation of equations (21) and (26).
HW 3 Due
Note: S4.1: how many initial conditions are needed for an nth order equation, meaning of fundamental set of solutions, linear dependence and independence, example 2, the general solution to a nonhomogeneous equation; S4.2: how to solve the constant coefficient nth-order linear homogenous problem, how to find the nth roots of -1.
Oct 5
Lab 2 Due
Note: S4.3: how MUC is different for higher-order equations (as compared to 2nd-order), why the correct guess in example 1 is yp = t3et, similarly, how the guesses are obtained for examples 2 & 3.
Oct 7
Proj 1 Due
Note: S5.1: how a power series solution is similar to the MUC, the ten results about power series, how to shift the index of summation in a power series; S5.2: meaning of an ordinary and singular point, the form of the series solution at an ordinary point, how to find the coefficients of the series solution, what a recurrence relation is, how we must write the coefficients of P(x), Q(x) and R(x) to find a series solution around x0, how many terms we have to find in the series solution.
HW 4 Due
Note: S5.3: what statement we are justifying in this section, what we need to be true about p and q for this to work, meaning of ordinary and singular point in this context, what the radius of convergence of a series solution at an ordinary point will be, the radius of convergence of Q/P.
Oct 12
Note: S5.4: what an Euler equation is, how solutions are obtained, how solutions are obtained for repeated and complex roots, why we are unable to just ignore singular points, what a regular singular point is.
Oct 14
Note: S5.5: how we rewrite equation (1) to get equations (3) and (4), the form of the series solution we look for in the case of a regular singular point, how in example 1 a solution is obtained from this form.
Oct 18: Fall Break
(no class)
Oct 19: Fall Break
(no class)
HW 5 Due
Note: S6.1: what the comparison theorem for improper integrals says, what an integral transform is, what the Laplace transform is, the three steps to using the Laplace transform, how the transforms of f(t) = 1, eat, and sin(t) are obtained; S6.2: what the Laplace transform of f'(t), f''(t), etc., are, how we can transform a differential equation and find its solution by inverting the transform, the advantages of the transform method, how to use a table of transforms to solve differential equations.
Oct 21
Note: S6.3: what functions appear in some of the more interesting applications of the Laplace transform, what the unit step function is, the transform of uc(t) and uc(tf(t-c), theorem 6.3.2; S6.4: the solution of examples 1 and 2.
Oct 25: Read 6.5; Lab 3
HW 6 Due
Note: S6.5: what an impulse, I(t) is, and how it is related to the forcing term g(t) in an equation, how we use a limiting process to define the unit impulse function δ, what we get when integrating δ(t-t0)f(t) over an interval containing t0.
Oct 26 Oct 27: [Review] Oct 28 Oct 29: Midterm
Sections 1.1–5.5
HW 7 Due
Note: S6.6: what L-1{F(s)G(s) is, the definition of the convolution (f *g)(t), properties of the convolution operator, how the convolution theorem is used in examples 1 and 2, what the transfer function for a differential equation is.
Nov 2
Lab 3 Due
Note: S7.1: how a second-order equation is transformed into a system of two first-order equations (ex. 1), what a solution to a system is, how thm. 7.1.1 is similar to thm. 2.4.2, how thm. 7.1.2 is similar to thm. 2.4.1; S7.2: how to multiply matrices and vectors, what invertible and singular matrices are; S7.3: when solutions to Ax = 0 exist, and how many there are, what eigenvalues and eigenvectors are, the solutions of example 4.
Nov 4
Note: S7.4: how superposition works for systems, what a general solution to a system is, a fundamental solution set, how the Wronskian works with this.
Proj 2 Due
Note: S7.5: what a phase plane and phase portrait are, how two solutions are found for example 1, how in example 1 the line x2 = 2 x1 is found, how trajectories on this line are found, how the phase portrait is then obtained, what a saddle point and node are, the general theory for finding solutions using eigenvalues and eigenvectors.
Nov 9
Note: S7.6: how the real-valued solutions are found from a complex valued solution in example 1, what a spiral point is, what cases in solving systems there are beyond the three on p.404.
Nov 11
HW 8 Due
Note: S7.7: what a fundamental matrix is, what Ψ(t) (Psi(t)) and Φ(t) (Phi(t)) are and how they are related, how exp(At) is related to eat, what an uncoupled system is, how a matrix A may be diagonalized, how this is used in example 3 and following to solve x' = A x.

Note: S7.8: what additional solutions to the homogeneous system looks like when there are repeated eigenvalues with distinct eigenvectors, and with non-distinct eigenvectors, what a generalized eigenvector is and how it is related to this problem.
Nov 16
Note: S7.9: how the transformation x = Ty simplifies the nonhomogeneous system, when undetermined coefficients can be used for a nonhomogeneous system, how this differs from the case of a single equation, how variation of parameters is more general than diagonalization or undetermined coefficients, what the (matrix) equation satisfied by u in variation parameters is, how this is solved in example 3, how Laplace transforms are used in example 4.
Nov 18 Nov 19: Review
HW 9 Due
Note: S8.1: what the Euler method is, how it is related to a difference quotient, what the backward Euler formula is, what we mean by convergence of a numerical method, what truncation and round-off errors are, how big the global truncation error is for the Euler method; S8.2: what the improved Euler formula is, how the error in the formula depends on step size, how we might vary step size in a calculation to improve accuracy.
Nov 23
Lab 4
Note: S8.3: what the Runge-Kutta method is, how its error depends on h.
Nov 25: Thanksgiving Nov 26: Thanksgiving
(no class)
Note: S9.1: what understanding the methods of chp 9 give of solutions to nonlinear systems, what equilibrium solutions and critical points are, what the phase plane and a phase portrait are, how the phase portraits in cases 1–5 are obtained, how the conclusions on p.493 and in the table on p.494 reflect the different eigenvalues found in cases 1–5.
Nov 30
Note: S9.2: the significance of autonomous systems for our analysis and trajectories in the phase plane, the meaning of stable, unstable, and asymptotically stable critical points, why critical points are interesting when the behavior of solutions, what a basin of attraction and separatrix are, how we (might) be able to solve to find the equations of trajectories in the phase plane.
Dec 2
Lab 4 Due
Note: S9.3: why stability is of interest in physical systems, how small perturbations to A may change its eigenvalues, what we mean by a locally linear system, how the system (10) may be determined to be locally linear, what the linearization of (10) is, what the Jacobian matrix is, why the linear and locally linear columns of the table on p.531 differ as they do.
HW 10 Due
Note: S9.4: how the systems in examples 1 and 2 are analyzed, what a nullcline is.
Dec 7 Dec 8: Lab 5 Dec 9 Dec 10: Review
Proj 3 Due
Dec 13: Lab 5 Due
Last class day
Dec 14 Dec 15: Final
1:30–3:30pm
Dec 16 Dec 17

Ma316-001-F10 Calendar