Spring 2009

Group 120501.01 - Prof. Dana-Picard

**Teacher:** Prof. Noach Dana-Picard

**Teacher Assistents:**
Mr Y.O. Koch and Mr A. Yarden

**Course coordinator:** Dr V. Lender.

**Syllabus:**

- What is an ordinary differential equation? Examples from Physics, from Geometry. Slope fields.
- Equations with initial conditions.
- Solutions of an ordinary differential equation. General solution, particular solution, singular solution.
- First order equation, separable equation, exact equation, integrating factor.
- Homogeneous linear equation of first order. Non homogeneous linear equation of first order: the method of undetermined coefficients, and the method of variation of parameters. Bernoulli equations, Clairaut equations.
- Second order linear equations with constant coefficients: the homogeneous case. Reduction of order.
- Second order linear equations with constant coefficients - the non-homogeneous case: undetermined coefficients and variation of parameters.
- Cauchy-Euler equations.
- Higher order linear equations.
- The exponential of a square matrix.
- Systems of linear differential equations of first order. Matrix methods.
- Laplace transform; inverse Laplace transform. Solution of differential equations using Laplace transform.
- Solution of systems of differential equations using Laplace transform.
- Series solutions of differential equations.

- Have a look at some vector fields and solutions of ODE .
- Introduction: pdf, pps.
- Slope fields: pps.
- Euler method: pdf.
- existence and Uniqueness: pdf.
- Linear ODEs of high order: pdf.
- Systems of differential equations of first order: pdf, pps.
- Laplace Transforms: pdf,presentation.

And here is a pdf file for the first lecture.

- Non homeogeneous linear equation with constant coefficients; method of Variation of Constants: ps; pdf.
- Linear equation with the annihilator method: ps; pdf
- A non-homogeneous Cauchy-Euler equation:page 1 , page 2 .
- Linear equation with Laplace Transform: ps; pdf.
- Targil 7 question 5b.
- Targil 7 question 5-vav page 1 - page 2 .
- Targil 7 question 6.
- Targil 12.

**Former Exams:**

- Spring Term 2001: Moed A - moed B.
- Spring Term 2003: Moed A - moed B.
- Solutions of Spring Term 2004 (first session): 1, 2, 3, 4, 5, 6, 7.

**Recommended Texts:**

- Nagle and Saff: Fundamentals of Differential Equations, Addison-Wesley.
- Abell and Braselton: Modern Differential Equations, Saunders College Publishing.
- Differential Equations, Schaum Series (exists in Hebrew and in English).
- G. Simmons and S. Krantz: Differential Equations: Theory, Technique, and Practice, MacGraw-Hill, 2007.

All these books are in the library. We recommend a visit to the library; there are many recent acquisitions.

In Calculus books, there is generally a chapter on Differential Equations, but this not enough for this course.

**Useful links:**

- We recommend strongly the usage of the online exercises, plotters and function calculators of the wonderful site WIMS , by Prof. Gang Xiao at the University of Nice (France).
- The Mathematical Atlas.
- The Math Archive.
- The Computational Science Education Project, ODE chapter.
- The ODE chapter of SOS Mathematics ; very nice and useful.
- Math Art Gallery; have a look at this.