# Math 246 Description and Prerequisites

## UM Undergraduate Catalog Course Description

Math 246, Differential Equations for Scientists and Engineers (3 credits) An introduction to the basic methods of solving ordinary differential equations. Equations of first and second order, linear differential equations, Laplace transforms, numerical methods, and the qualitative theory of differential equations.

## Course Prerequisites

• Math 141 (Calculus II) or its equivalent;
• Either Math 240 (Calculus III) or ENES102 or PHYS161 or PHYS171 or some other course with an adequate coverage of vectors.

## Our More Detailed Outline

Introduction to and Classification of Differential Equations
First-Order Equations
Linear, separable and exact equations
Introduction to symbolic solutions using MATLAB
Existence and uniqueness of solutions
Properties of nonlinear vs. linear equations
Qualitative methods for autonomous equations
Plotting direction fields using MATLAB
Models and applications
Numerical Methods
Introduction to a numerical solver in MATLAB
Elementary numerical methods: Euler, Runge-Trapezoidal, Runge-Midpoint, Runge-Kutta
Local and global error, reliability of numerical methods
Higher-Order Linear Equations
General theory of linear equations
Reduction of order
Homogeneous linear equations with constant coefficients
Methods of undetermined coefficients, key identity evaluations, variation of parameters, and Green functions for nonhomogeneous equations
Symbolic and numerical solutions using MATLAB
Mechanical vibrations
Laplace Transforms
Definition and calculation of Laplace transforms
Applications to differential equations with piecewise continuous forcing functions
First-Order Linear Systems
General theory
Systems with constant coefficients and matrix exponentials
Eigenpairs and special solutions
Finding eigenpairs and solving linear systems with MATLAB
The phase plane and parametric plotting with MATLAB
First-Order Systems in the Plane
Autonomous systems and stationary solutions
Linearized stability and phase plane analysis
Linearized stability analysis and plotting vector fields using MATLAB
Numerical solutions and phase portraits of nonlinear systems using MATLAB
Models and applications