| General Info | Calendar | Announcements | Additional Help | Homework | Matlab Homework | Practice Exams | Lecture Summaries|
General Information
Announcements & DatesImportant Dates and Class Holidays:
Final Exam InformationThe final exam is Friday, December 9, 8-11am. You can bring a "cheat sheet" which should be at most a page. Topics for the final. Practice exams can be found at here. Solutions to John Hall's practice final: page 1 , 2 , 3 , 4 , 5ab , 5cd , 6 , 7 , 8a, 8b, 8c . Dragos's office ours for final exam: Tue 4-5, Wed 3-5 in APM6101. Chris's office hours: Tue 1-3, Wed 1-3, APM 6446. Allison will be away this week, but if you are enrolled in her section, you can pick up your old homeworks from Chris. Midterm II InformationThe midterm is Friday, November 21, 12-12:50 in WLH 2005. You are allowed to have a "cheat sheet" which should be at most a page (front only). Review for Midterm II will be on Wednesday during lectures. Bring questions! Practice midterms can be found at here. The midterm topics. Midterm I InformationThe midterm is Wednesday, October 22, 12-12:50. You are allowed to have a "cheat sheet" which should be at most half a page back and forth. The midterm topics. We will take the midterm in 3 different rooms as follows:
Interactive campus map . Because of the midterm, the 4th Matlab is due on Friday at 3:50PM. Review for Midterm I will be on Monday during lectures. Bring questions! Practice midterms can be found at here. Additional HelpIf you are having trouble with the homework or have questions about the material, the best way to get help is to attend the office hours offered by me and the teaching assistants. If you can't make the scheduled times, then email us and we'll set up an appointment.Additional help is given by
Lecture SummariesLecture 1: Introduction to differential equations. Savings account modelling. Classification of differential equations: ordinary/partial, linear/nonlinear, order of an ODE. Lecture 2: Direction fields. Integral curves. Equilibrium solution. I solved the equation dy/dx=1/10 y-1 geometrically, showed that solutions diverge from the equilibrium solution. I also solved the equation analytically by separating variables and integrating, and analyzed the behaviour of solutions. Lecture 3: Linear 1st order ODE. Standard linear form. Integrating factors. I solved a couple of examples via integrating factors. Lecture 4: Modelling mixture problems. Existence and uniqueness of solutions for linear first order ODEs. The case of continuous vs. discontinuous coefficients. Integral curves don't cross. Lecture 5: Nonlinear 1st order ODEs. Separation of variables. Solutions may not be unique and they may have discontinuities which are not visible from the original equation. Lecture 6: Autonomous ODEs. Critical points. Phase line. Types of critical points: asymptotically stable, unstable, semistable. Example: population growth, logistic growth. Lecture 7: Exact 1st order differential equations: M+Ny'=0. How to check an equation is exact (M_y=N_x). Finding the function whose derivatives are M and N (e.g. f_x=M, f_y=N). This function f is constant along solutions. Examples. Lecture 8: Linear second order ODE and IVP. Superposition of solutions. Existence and uniqueness of solutions. The case of constant coefficients. The characteristic equation. Examples. Lecture 9: Wronskian for two solutions (y_1, y_2) is given as a 2 x 2 determinant. Nonzero wronskian means fundamental set of solutions (which means that the general solution is of the form y=c_1y_1+c_2 y_2). Abel's theorem shows how to compute the Wronskian. Lecture 10: Constant coefficient 2nd order ODE. The case of complex roots. Complex exponentials. The fundamental pair is obtained by taking the real and the imaginary part of the exponential. Lecture 11: Midterm review. I announced the midterm rooms, midterm topics, solved a few sample problems. Lecture 12: The case of repeated roots r_1=r_2=a. A fundamental pair of solutions is y_1=e^{at}, y_2=te^{at}. Non-homogeneous equations have solutions y=y_p+y_h where y_p is a particular solution and y_h solves the homogeneous equation. I showed an example where the particular solution is found by method of undetermined coefficients (using polynomials). Lecture 13: More undetermined coefficients. I showed the examples involving trigonometric functions, exponentials, and combinations of such. Lecture 14: Variation of parameters. I showed how to look for solutions of the form y=u_1 y_1+u_2 y_2, where y_1, y_2 solve the homogeneous equation and u_1, u_2 are functions to be determined. I found the expression of u_1, u_2. Examples. Lecture 15: Linear first order systems of ODEs. I showed how a first order linear system becomes a second order differetial equation and conversely. I solved a system which came from modelling temperature. Matrices: multiplication, multiplication by vectors, inverses. Lecture 16: Using matrices to solve systems. Homogeneous systems admit nontrivial solutions if the matrix of coefficients has zero determinant. Eigenvalues and eigenvectors. Eigenvalues are found by solving det(A-lambda I)=0. Lecture 17: Eigenvalues are found by solving the characteristic equation. Linear independence and basis. Check for basis by computing the determinant of the matrix whose columns are the given vectors. Lecture 18: Solving first order systems of ODEs X'=AX. Find two solutions in the form e^{lambda t} v, where lambda is an eigenvalue of A, and v is an eigenvector. Superposition of solutions. Linear independence and Wronskian. Sketching solutions: saddle (eigenvalues of opposite signs), sink nodes (negative eigenvalues), source nodes (positive eigenvalues). Lecture 19: Complex eigenvalues. Take the real and imaginary parts of the complex solutions. The trajectories are spirals. To find the direction (clockwise vs. counterclockwise) pick a point on the spiral and find the velocity vector. I started repeated eigenvalues. Lecture 20: Repeated eigenvalues and improper nodes. To find the direction of the trajectories, compute the velocity of one vector at a point. Fundamental matrix Psi(t) and the normalized fundamental matrix Phi(t)=Psi(t) Psi(0)^-1. Solving IVP: X=Phi(t)X_0. Lecture 21: Exponential of a matrix. The normalized fundamental matrix is Phi(t)=e^{At}. Diagonalization. Lecture 22: Diagonalization continued T^{-1}AT=D, D=diagonal with eigenvalues on the diagonal. Two applications: decoupling systems (even when they are not homogeneous) x'=Ax+b. Look for x=Ty, then y'=Dy+T^{-1}b. Second application: compute exponentials e^{At}=Te^{Dt}T^{-1}. Lecture 22: Midterm review. Lecture 23: Radius of convergence of power series. Series solutions near ordinary points. I solved y''+y=0, found the recursion for the series coefficients. Lecture 24: Laplace transform. Examples: Laplace transform of 1, e^{at}, sin (at), cos(at), t^n. Inverse Laplace transform. Partial fractions. Lecture 25: Laplace transform of derivatives. Solving ODEs using Laplace transform. Lecture 26: Step functions. Laplace transform of step functions. The inverse Laplace transform of e^{-sa}F(s) is u_a(t)f(t-a). I solved a differential equation involving step functions. |
