# Math 20D - Introduction to Differential Equations - Fall 2011

| General Info | Calendar | Announcements | Additional Help | Homework | Matlab Homework | Exams | Lecture Summaries|

## Announcements & Dates

Important Dates and Class Holidays:
• Friday, September 23: First lecture.
• Friday, October 21: Midterm I
• Friday, October 21: Drop deadline
• Friday, November 11: Veterans' Day
• Friday, November 18: Midterm II
• Thursday-Friday November 24-25: Thanksgiving Recess -- No Class
• Thursday, December 1: Matlab Quiz
• Friday, December 9: FINAL EXAM, 8-11am

## Midterm I

Fall 2008 Midterm I - Solutions.

Midterm I Topics

## Midterm II

Midterm II Topics

Fall 2008 Midterm II - Solutions.

## Final Exam

Final Exam Topics

Fall 2008 Final Exam - Solutions.

If 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 Summaries

Lecture 1: Course introduction. Outline. Classification of differential equations: linear/nonlinear, first order etc.

Lecture 2: Geometric methods. Direction fields, integral curves. Integral curves do not cross or touch. Examples.

Lecture 3: Constant coefficient first order linear equations. Equilibrium solutions. Variable coefficient first order linear equations. Method of integrating factors. Standard linear form.

Lecture 4: More on integrating factors. Examples. Existence and uniqueness of solutions. Discontinuities of coefficients and domain of definition of solutions.

Lecture 5: Separable equations. Examples. Modeling with differential equations: mixture problems.

Lecture 6: Autonomous equations. Criticial points. Phase portrait. Classification of critical points: asymptotically stable, semistable, unstable. Logistic growth.

Lecture 7: Exact differential equations: M+Ny'=0 where M=f_x and N=f_y. Check for exactness: M_y=N_x. Finding the potential function f. The potential function is constant along solutions: f(x,y)=c.

Lecture 8: Second order constant coefficient differential equations and IVP. Characteristic equation. Case of real roots. General solution is found by superposition.

Lecture 9: Case of complex roots. Complex exponentials. Real valued solutions are found by taking real and imaginary part of the complex valued solutions. Oscillations. Case of repeated roots.

Lecture 10: General theory of linear second order homogeneous equations. Superposition of solutions. Wronskian is given by a 2 x 2 determinant. Fundamental pairs of solutions have non-zero Wronskian. Abel's theorem.

Lecture 11: General theory of inhomogeneous equations. Solutions are of the form y=y_p+y_h. Finding the particular solution y_p by undertmined coefficients. The exponential case: x_p=e^{at}/p(a). When exponent is a root of the characteristic equation, x_p=te^{at}/p'(a).

Lecture 12: Undetermined coefficients: polynomial, trigonometric and mixed cases. Midterm review.

Lecture 13: Variation of parameters and examples. 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 determined u_1, u_2.

Lecture 14: Systems of first order linear equations. I solved an explicit system. I showed how a first order linear system becomes a second order differetial equation and conversely.

Lecture 15: Matrices and vectors. Products. Determinants. Invertible matrices. Calculating inverses. Solving systems of linear equations. Eigenvalues. Eigenvectors. Characteristic polynomials.

Lecture 16: Linear independence. Span. Basis. General theory of first order systems. Solutions of the form e^{\lambda t} v, where lambda is an eigenvalue and v is an eigenvector. Superposition of solutions. Fundamental set of solutions and the Wronskian.

Lecture 17: Solving homogeneous systems with constant coefficients by finding eigenvalues and eigenvectors. Real distinct eigenvalues: origin can be a saddle (eigenvalues of opposite signs) or node (eigenvalues of the same sign). Unstable or asymptotically stable nodes.

Lecture 18: Complex eigenvalues and spirals. To find real valued solutions, take real and imaginary part of complex valued solutions. Finding the direction of the spiral by computing velocity vectors. Repeated eigenvalues. Finding the fundamental pair of solutions by undetermined coefficients.

Lecture 19: Sketching improper nodes. I showed a diagram summarizing the types of trajectories. I discussed the exceptional cases: zero eigenvalues or purely imaginary eigenvalues.

Lecture 20: Fundamental matrices. Normalized fundamental matrix Phi(t)=Psi(t)Psi(0)^{-1}. Solving initial value problems x=Phi(t) x_0. Exponentials of matrices e^{At}=Psi(t)Psi(0)^{-1}=Phi(t).

Lecture 21: Inhomogeneous systems. Undetermined coefficients. Variation of parameters for systems x'=Ax+g(t). Particular solution given by x_p=Psi(t) integral Psi(t)^{-1} g(t).

Lecture 22: Series methods. Series. Radius of convergence. Taylor series. Examples. Solving differential equations using series: finding recursive formulas between coefficients.

Lecture 23: Laplace transform. Functions of exponential growth. I calculated the Laplace transform of 1, e^{at}, cos (at), sin (at) and t^n.

Lecture 24: Shift rule: Laplace of e^{at} f(t) is F(s-a). Laplace transforms of derivatives. Inverse Laplace transform and partial fractions. Using Laplace transform to solve initial value problems.

Lecture 25: 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.

Lecture 26: Final Exam Review.