A Complete First Course in Differential Equations

• Classify differential equations according to their type and order.
• Solve first order differential equations that are separable, linear, homogeneous, exact, as well as other types that can be solved through different substitutions.
• Use first order differential equations to model different applications from science.
• Solve linear second order equations with constant coefficients (both homogenous and non-homogeneous) using the method of undetermined coefficients, variation of parameters, and Laplace transforms.
• Understand the theory of linear second order differential equations and how it relates to ideas from linear algebra.
• Use linear second order equations with constant coefficients (both homogenous and non-homogeneous) to model applications from science.
• Find Laplace and inverse Laplace transforms.
• Use Laplace transforms to solve linear second order equations with constant coefficients which contain forcing functions such as impulses, step functions, and periodic functions.
• Solve systems of linear differential equations with constant coefficients and understand the importance of eigenvalues and eigenvectors for finding solutions.
• Understand the importance of the Matrix exponential and how to compute it in order to find the solutions of linear systems of differential equations.
• Apply basic numerical methods to find approximate solutions of differential equations.
• Understand the basics of some complex analysis and its usefulness to differential equations.
• Use equilibrium points, phase portraits, and stability analysis to analyze linear systems.
• Use Maple to analytically and numerically solve differential equations. Use Maple to study differential equations qualitatively.
• Model real world phenomenon with differential equations.
• Find series solutions to second order linear equations with variable coefficients. Apply this method to ordinary points and regular singular points. Find Frobenius series solutions using the method of Frobenius. Apply reduction of order to find series solutions.
• Use Fourier series to solve partial differential equations. Solve the heat, wave, and Laplace equation using separation of variables and Fourier Series. Understand theory and applications of General Fourier series, Sine Fourier series, Cosine Fourier series, and convergence of Fourier series. Solve inhomogenous PDEs.
• Use theory of vector spaces, orthogonality of functions and inner products, self adjoint operators and apply to Sturm-Liouville Eigenvalue problems. Use eigen function expansions to solve nonhomogenous problems.
• Analyze nonlinear autonomous system by finding equilibrium points and stability. Understand concept of linearization and the Hartman-Grobman Theorem. Find and analyze Hopf bifurcation as well as other commonly known bifurcations
• Apply Numerical methods and understand importance of stability and accuracy. Be able to implement in Maple. Be able to use state of the art DE solvers.

This course will teach everything that is usually taught in the first two semesters of a university/college course in differential equations. The topics we will consider in this course are

1. First Order Differential Equations
2. Linear Equations of Higher Order
3. Laplace Transform Methods
4. Linear Systems of Differential Equations
5. Power Series Methods
6. Partial Differential Equations
7. Fourier Series
8. Sturm Liouville Eigenvalue Problems
9. Nonlinear Systems of Differential Equations
10. Numerical Methods