Analytic solutions of first-order and linear second-order differential equations are covered. Eigenvalue methods are employed to solve systems of homogeneous differential equations and extended to the inhomogeneous case. Series solutions to higher-order differential equations with non-constant coefficients and Laplace transform methods for initial value problems are developed. Partial differential equations are introduced and solved using Fourier integral methods under a variety of boundary conditions. Vector functions and their associated operations are introduced, starting with differential operations (i.e. curl and divergence) and then integral operations (i.e. line integrals, surface integrals) including Stokes', Green's and Divergence theorems. The emphasis of the material is on its use in areas such as fluid flow and mechanics of solids. The laboratory component of the course emphasizes engineering applications in mechanical vibrations, control systems, robotics, fluid dynamics, material mechanics and system modelling. A computing tool such as Maple or MatLab is used throughout.
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Upon successful completion of this course, the student will be able to:
Model physical processes using higher-order differential equations. [1, 2]
Use eigenvalue methods to analytically solve homogeneous higher-order linear differential equations with constant coefficients and use the methods of undetermined coefficients and variation of parameters to extend this to the inhomogeneous case. [1, 2]
Analytically solve first- and second-order linear systems of differential equations in both the homogeneous and inhomogeneous cases. 
Solve higher-order linear differential equations using series methods about ordinary and regular singular points. [1, 2]
Use Laplace transform methods, including partial fractions and convolutions, to solve differential equations expressing initial value problems. [1, 2]
Use Fourier integral methods to solve initial and boundary value problems involving separable partial differential equations. [1, 2]
Apply vector differential and integral operations to vector fields, such as in fluid flow, solid mechanics and electric fields, and to scalar fields such as fluid density. 
Interpret the physical meaning of curl and divergence operations when applied to the dynamics of fluid flow. 
Distinguish between conservative and non-conservative fields and the implication for calculations. 
Use vector theorems - Gauss' Divergence, Green's and Stokes' theorems - to transform between different vector calculus calculations in order to solve most efficiently. [1, 2]
Demonstrate proficiency with the mathematics for fluid dynamics and mechanics of solids. [1, 2]
Effective as of Fall 2018
MATH 4499 is offered as a part of the following programs:
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