21-260 Differential Equations

Description

The subject of differential equations can be described as the study of equations involving derivatives. It can also be described as the study of anything that changes. The reason for this goes back to differential calculus, where one learns that the derivative of a function describes the rate of change of the function. Thus any quantity that varies can be described by an equation involving its derivative, whether the quantity is a position, velocity, temperature, population or volume.

There are three main ways to study differential equations. There are analytic methods, wherein a mathematical formula for a solution of a differential equation is obtained. There are Numerical techniques, which provide an approximate solution, generally using a computer or programmable calculator. Differential Equations can also be studied qualitatively, determining general properties of solution without concern for exact behavior.

In this course, we will emphasize analytic methods, though qualitative and numerical techniques will make brief appearances.

Learning Objectives

After completing this course, you should be able to

1. Know what is meant by a "differential equation."
2. Determine if a given function is a solution to a particular differential equation.
3. Understand how the terms linear, non-linear, order, ordinary and partial are used to classify differential equations.
4. Find all solutions of a separable differential equation.
5. Find the general solution to a linear first order differential equation.
6. Use basic mathematical models to describe physical situations.
7. Apply the theorems for existence and uniqueness of solutions to differential equations appropriately.
8. Model the growth or decay of populations using autonomous differential equations.
9. Use vectors and matrices to solve linear systems of algebraic euqations.
10. Determine whether or not a set of vectors is linearly independent.
11. Find the eigenvalues and eigenvectors of a matrix.
12. Find the general solution for a first order, linear, constant coefficient, homogeneous system of differential equations.
13. Sketch and interpret phase plane diagrams for systems of differential equations.
14. Compute solutions to second order, liner, constant coefficient, homogeneous differential equations.
15. Use the method of undetermined coefficients to solve second order, liner, constant coefficient, non-homogeneous differential equations.
16. Use second order, linear, constant coefficient differential equations to model physical situations, including mechanical vibrations.
17. Compute the Laplace transform of a function using the definition.
18. Apply a variety of theorems to determine Laplace transforms and inverse Laplace transforms.
19. Use Laplace transforms to compute solutions to linear, second order differential equations.
20. Find the Fourier series of a periodic function.
21. Find the Fourier sine and cosine series for a function defined on an interval.
22. Use the method of separation of variables to find solutions to some partial differential equations.
23. Find solutions to the Heat Equation and the Wave Equation for a variety of initial data.

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